current address:]Department of Photonics Engineering, DTU Fotonik, Technical
University of Denmark, Building 343, DK-2800 Kgs. Lyngby, Denmark current
address:]Departament de Física Aplicada, Universitat de Barcelona, IN2UB,
08028 Barcelona, Spain current address: ]Instituto de Ciencia de Materiales de
Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC), Calle
Sor Juana Inés de la Cruz, 3, 28049 Madrid, Spain.

# Engineering nanoscale hypersonic phonon transport

O. Florez<EMAIL_ADDRESS>Catalan Institute of Nanoscience and
Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona,
Spain  Dept. de Física, Universitat Autonoma de Barcelona, 08193 Bellaterra,
Spain  G. Arregui  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain  [  M.
Albrechtsen  Department of Photonics Engineering, DTU Fotonik, Technical
University of Denmark, Building 343, DK-2800 Kgs. Lyngby, Denmark  R. C. Ng
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST,
Campus UAB, Bellaterra, 08193 Barcelona, Spain  J. Gomis-Bresco  Catalan
Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB,
Bellaterra, 08193 Barcelona, Spain  [  S. Stobbe  Department of Photonics
Engineering, DTU Fotonik, Technical University of Denmark, Building 343,
DK-2800 Kgs. Lyngby, Denmark  NanoPhoton - Center for Nanophotonics, Technical
University of Denmark, Ørsteds Plads 345A, DK-2800 Kgs. Lyngby, Denmark  C. M.
Sotomayor-Torres  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain  ICREA -
Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain  P.
D. García<EMAIL_ADDRESS>[  Catalan Institute of Nanoscience and
Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona,
Spain

###### Abstract

Controlling the vibrations in solids is crucial to tailor their mechanical
properties and their interaction with light. Thermal vibrations represent a
source of noise and dephasing for many physical processes at the quantum
level. One strategy to avoid these vibrations is to structure a solid such
that it possesses a phononic stop band, i.e., a frequency range over which
there are no available mechanical modes. Here, we demonstrate the complete
absence of mechanical vibrations at room temperature over a broad spectral
window, with a 5.3 GHz wide band gap centered at 8.4 GHz in a patterned
silicon nanostructure membrane measured using Brillouin light scattering
spectroscopy. By constructing a line-defect waveguide, we directly measure GHz
localized modes at room temperature. Our experimental results of thermally
excited guided mechanical modes at GHz frequencies provides an efficient
platform for photon-phonon integration with applications in optomechanics and
signal processing transduction.

###### pacs:

(42.25.Dd, 62.25.-g, 46.65.+g, 42.50.Wk)

Nanostructured materials offer the possibility to manipulate the mechanical
vibrations of a solid over a specified spectral bandwidth. This in turn
enables the control of light-matter interactions in the visible and near-
infrared regimes for optomechanical applications ranging from high-resolution
accelerometers [1] to mass and force sensors [2, 3], in addition to providing
fundamental insight into phenomena such as quantum ground-state cooling [4,
5]. By periodically distributing the mass within a system, it is possible to
engineer its mechanical modes [6, 7] and open frequency windows over which the
destructive interference of scattered waves forbids any phonon propagation [8,
9]. This approach enables engineering of the thermal conductance of the
structure [10] and allows for the routing of phonons at the mesoscale [11,
12]. Although full-gap GHz phononic crystals are widely used in optomechanical
systems to create phononic shields [13], waveguides [14, 15], and cavities
[16, 17], clear and direct experimental evidence of a complete omnidirectional
phononic band gap at hypersonic (GHz) frequencies is still lacking. Existing
experimental work is generally limited to MHz frequencies, using piezoelectric
materials to drive the system [18, 19, 20], requiring varying interdigitated
electrodes to probe different frequencies and propagation directions. In the
GHz regime, only partial and narrow mechanical band gaps (with up to $8{\%}$
gap to mid-gap ratio) have been shown using assembled platforms such as
colloidal crystals [21] or two-dimensional phononic crystal membranes [22].
Furthermore, the control and guiding of mechanical waves at GHz frequencies
has been difficult to achieve or measure, relying on complex optomechanical
systems [14, 15], or nonlinear stimulated phenomena [23].

Here, we report direct experimental evidence of a wide full phononic gap with
a central frequency at 8.4 GHz and a spectral width of 5.3 GHz (a gap to mid-
gap ratio of 64%) in a free-standing patterned silicon membrane phononic
crystal. Additionally, we create a line-defect waveguide with the same
geometry in which we directly measure two guided modes at 5.7 and 7.1 GHz
within the band gap at room temperature. We demonstrate the spectral
tunability of the mechanical gap from approximately 4 to 12 GHz, which
subsequently enables spectral tunability of the guided modes. All the
structures measured here are fabricated on a silicon-on-insulator (SOI)
platform, which readily enables integration with electronic and photonic
circuits. Figure 1(a) shows a scanning electron micrograph (SEM) of the
fabricated pattern composed of a triangular array of “shamrocks” [24, 25],
formed by three tangential circles with nominal parameters of thickness
$t=220$ nm, period $a=330$ nm, and radius $r=0.22a$, as detailed in the inset
of Fig. 1(a) and Fig. S1 in the supplementary information (SI). We calculate
the mechanical dispersion relation of the structure by solving the full three-
dimensional elastic wave equation using finite-element (FEM) simulations
performed with COMSOL Multiphysics [26]. Figure 1(b) plots the symmetric
(blue) and asymmetric (red) acoustic modes with respect to the mid-plane of
the silicon slab, calculated over the entire first Brillouin zone (BZ) of the
crystal. We use the geometrical parameters extracted from SEM images to more
accurately simulate the real shape of the fabricated crystal (see Fig. S3 in
SI for a statistical analysis). We also take into account the anisotropy of
the silicon stiffness tensor and its particular orientation with respect to
the fabricated samples, as detailed in Fig. 1(a). Due to this mechanical
anisotropy, the irreducible BZ is determined by the first quarter of the
hexagon highlighted on the bottom part of Fig. 1(b) (see section S3 in SI). A
full mechanical gap opens between the 6th and 7th bands, from 6.7 GHz up to
11.4 GHz (gap to mid-gap ratio of 52 %) which results in the complete
depletion of the phonon density of states (DOS) over this frequency range, as
shown in Fig. 1(c). The particular shape of the shamrock crystal, which is
comprised of large masses connected by small necks, enables a distribution of
the mass within the unit cell that results in this broad mechanical gap. A
direct link exists between the spectral width of the gap and the narrow necks
(shorter distance between shamrocks): a larger radius leads to narrower
connected neck regions, which subsequently widens the gap [27].

Figure 1:  Shamrock phononic insulator. (a) Scanning-electron micrograph
(tilted-top view) of the fabricated structure on a silicon-on-insulator
substrate with a thickness of $t$ = 220 nm. The inset schematically
illustrates the geometrical parameters of the unit cell (highlighted in red)
with lattice constant $a$ = 330 nm, hole radius $r=0.22a$, and the distance
between the center of the shamrock and the center of each circle
$f=2r/\sqrt{3}$. (b) Simulated 3D mechanical dispersion relation of the
crystal over the first Brillouin zone. Blue and red curves indicate the
symmetric and asymmetric modes with respect to the middle plane of the silicon
slab at $t/2$. (c) Calculated phononic density of states (DOS) of the
structure. The light-blue region highlights the full mechanical gap spanning
6.7 GHz to 11.4 GHz. Figure 2:  Brillouin light scattering spectroscopy. (a)
Schematic illustration of Brillouin scattering with the phase-matching
condition for the backward configuration used in the experiments. Here,
$\vec{k}_{i}$ and $\vec{k}_{s}$ represent the incident and the scattered
light, respectively, and $\vec{q}_{\parallel}$ is the parallel mechanic
wavevector. The magnitude of $\vec{q}_{\parallel}$ depends on the incident
angle, where $q_{\parallel}=2k_{i}\sin{\theta}$. (b) Measured Brillouin
scattering spectrum for an incident angle of $\theta=32.5^{\circ}$ with
p-polarized light. The green central peak stems from elastic Rayleigh
scattering. Negative and positive frequency peaks on either side of this large
central peak correspond to Stokes and anti-Stokes contributions, respectively.
The light-blue regions highlight the mechanical gap. (c) Calculated dispersion
relation based on the geometrical parameters obtained from SEM images of the
fabricated samples that include a $4^{\circ}$ sidewall angle correction in the
vertical profile (inset). The black dots represent the measured frequencies of
vibrational modes for different angles and the vertical dotted line indicates
the frequencies obtained from the measured spectrum shown in (b). The
intensity color scale represents the normalized coupling coefficients for the
MB perturbation. (d), (e) and (f) indicate the direction in which the sample
is physically rotated to scan along the highest-symmetry directions $\Gamma
K$, $KM$, and $\Gamma M$, respectively. The green arrows indicate the
direction of the incident laser light while the other colored arrows
correspond to the rotation direction during measurements, which represent (and
are color-consistent with) the highest-symmetry direction indicated in (c).

We use Brillouin light-scattering spectroscopy [28, 29] to reconstruct the
mechanical dispersion relation of the system. For simplicity, we probe the
band structure along the $\Gamma KM\Gamma$ path, highlighted at the bottom of
Fig. 1(b), as the edges of the gap do not change in frequency with respect to
the irreducible BZ (see section S2 in SI). When incident light with frequency
$\nu_{i}$ and wavevector $\vec{k}_{i}$ reaches the surface of the sample with
a certain angle, $\theta$, as illustrated in Fig. 2(a), part of it is linearly
scattered while another small part is nonlinearly scattered in all directions
by thermally activated acoustic phonons. This scattering process occurs either
by the elasto-optic [30] or the moving-boundary (MB) [31] mechanism. The
former is a volumetric effect caused by the acoustic modulation of the
dielectric constant $\epsilon$ inside the material, while the latter is a
surface effect induced by the movement of phonons that creates corrugation at
the interface. The interplay between these two effects can result in the
enhancement [32] or the cancellation [33] of the scattering process. Given the
high refractive-index contrast between silicon and the surrounding air, and
the small volume of interaction, determined by the direction of the incident
beam and the thickness of the suspended structure, the scattering process here
is dominated by the MB mechanism (see S4 in the SI). Our experiment collects
the backscattered signal, $\vec{k}_{s}$. For this configuration, the phase-
matching condition for the mechanical wavevector $\vec{q}_{\parallel}$, which
lies parallel to the surface, is determined by

$q_{\parallel}=2k_{i}\sin\theta=\frac{4\pi}{\lambda_{i}}\sin\theta,$ (1)

where $k_{i}=2\pi/\lambda_{i}$. Therefore, it is possible to probe different
mechanical wavevectors by changing the angle of incidence of light $\theta$,
illustrated in Fig. 2(a), and subsequently map the dispersion relation of the
acoustic phonons. All measurements were taken by focusing a green laser
($\lambda_{i}$ = 532 nm) that is p-polarized with respect to the sagittal
plane formed by the angle $\theta$ (see Fig. S6 in SI). The scattered light
that was analyzed is also the p-polarized component. Although in-plane and
out-of-plane mechanical modes in bulk materials and membranes can be
selectively detected using light polarization [34], mechanical modes in
phononic crystals are generally mixed. Therefore, we do not obtain different
information by considering different polarizations of incident and analyzed
light.

Figure 3:  Hypersonic phononic waveguide (a) SEM image of a shamrock phononic
waveguide with a lattice period of $a$ = 440 nm and waveguide width of $w$ =
184 nm. The thickness and radius of the structure are the same as in previous
structures ($t=$ 220 nm, $r=0.22a$). (b) Measured Brillouin scattering spectra
in the waveguide (top) and surrounding phononic crystal (bottom) as is
illustrated in the insets, for an incident light angle of $23.8^{\circ}$. The
spectral width of the measured gap is indicated by the blue regions. Two peaks
whose frequencies correspond with the guided modes of the system appear inside
the gap in the spectrum of the waveguide (top). (c) Calculated dispersion
relation of the waveguide. The intensity color scale represents the normalized
coupling coefficient for the MB perturbation. The horizontal and vertical
dotted lines indicate the mechanical band edges and phononic wave vector,
respectively, while the black dots represent the frequencies of peaks 1 and 2,
all for the top waveguide spectrum shown in (b). The insets in (c) display the
mode profiles for the indicated bands where the color represents the
normalized out-of-plane displacement.

Figure 2(b) plots the mechanical spectrum measured with incident angle
$\theta=32.5^{\circ}$, which corresponds to the high-symmetry point $K$ in
reciprocal space. The central peak highlighted in green corresponds to the
elastic (Rayleigh) scattered signal. Positive and negative frequencies
correspond with anti-Stokes and Stokes contributions respectively, which are
equally likely in a stochastic process such as spontaneous Brillouin
scattering [30, 35]. All the peaks observed in the spectrum correspond to
vibrational modes of the system and their amplitudes depend on the scattering
efficiency of each mode with the incident laser light [36], which is
proportional to the displacement of the boundaries, as previously mentioned
and detailed in section S4 of the SI. We obtain the phonon frequencies by
fitting each of the observed peaks to Lorentzian line shapes and extracting
the mean value between the resonant frequencies of the Stokes and anti-Stokes
components. Figure 2(c) plots the mechanical dispersion relation along the
$\Gamma KM\Gamma$ path. The intensity color scale represents the normalized
coupling coefficients for the MB perturbation.

The sidewalls of our structures are angled at $4^{\circ}$ relative to
vertical, which is taken into account in our band structure calculation, as
shown in the inset of Fig. 2(c). This breaks the up-down symmetry and the
mechanical modes of the real structure therefore can not be classified by
their symmetry with respect to the mid plane of the slab as done previously in
Fig. 1(b). For this reason, all bands in Fig. 2(c) are indicated with the same
color and only change in intensity to indicate the scattering efficiency of
each mode and wavevector. We observe that this small correction to the
vertical profile causes a displacement of about 1 GHz in the bands below the
band gap which becomes evident upon comparing Fig. 2(c) and Fig. 1(c).
Additionally, the gap to mid-gap ratio increases from 52 % in Fig. 1(b) and
1(c), to 64 % in Fig. 2(c). The black dots in Fig. 2(c) are the measured
frequencies of the peaks as the incident angle is varied. The vertical dotted
line indicates the position of frequencies obtained for the spectrum shown in
Fig. 2(b).

To resolve the full mechanical dispersion relation, we map the highest-
symmetry directions of the Brillouin zone: $\Gamma K$, $KM$, and $\Gamma M$.
The $\Gamma K$ path is measured by varying the angle of incidence $\theta$
from zero to $32.5^{\circ}$, as depicted in Fig. 2(d) where the green arrow
represents the incident laser. Here, the value of the maximum angle $\theta$
is calculated from the relation $|\Gamma
K|=\frac{4\pi}{3a}=\frac{4\pi}{\lambda_{i}}\sin{\theta}$. To map the $\Gamma
M$ direction, we rotate the sample 30 degrees to align the $\Gamma M$ path
with the horizontal direction as indicated in Fig. 2(f) and, from that
position, we rotate the angle $\theta$. Here, the maximum angle is indicated
by $|\Gamma M|=\frac{2\pi}{\sqrt{3}a}=\frac{4\pi}{\lambda_{i}}\sin{\theta}$.
Mapping the $KM$ path requires the simultaneous variation of two specific
angles $\alpha$ and $\theta$ to measure the intersecting point of the blue
segment and the horizontal direction, as depicted in Fig. 2(e). The calculated
and measured frequencies are in good agreement and we attribute the residual
frequency mismatch to fabrication fluctuations and the non-vertical sidewalls.
Some modes in Fig. 2(c) are undetectable in the experiment as their
displacement is predominantly in-plane and therefore do not scatter enough
light to be detected. The light-blue region in Fig. 2(c) highlights the
mechanical gap of this particular crystal. Within this frequency window, no
mechanical modes were measured for any angle of incidence in any high-symmetry
direction, covering a broad spectral range of 5.3 GHz centered at 8.4 GHz,
which corresponds to a gap to mid-gap ratio of 64%. This crystal has the
largest measured mechanical gap in the hypersonic regime to date. We also
explore the possibility to spectrally tune the gap as a function of the
geometry by variation of the lattice constant $a$. The band gap evolution
calculated from FEM and measured spectra for crystals with periods of 220 nm,
330 nm and 440 nm, can be found in section S5 of SI. We confirm the spectral
tunability of the gap from 4 to 12 GHz. Subsequently, tuning of phononic
guided modes is also possible.

Finally, we demonstrate the possibility to create phononic waveguides with the
phononic insulator presented here. For this, we design and fabricate a
waveguide surrounded on both sides by shamrock phononic crystals with inverted
symmetry as shown in the SEM image in Fig. 3(a). This structure has a
periodicity of $a$ = 440 nm, a waveguide width of $w$ = 184 nm, and the same
fill fraction and thickness as the previous structures ($r/a=0.22$, $t=220$
nm). The mirror symmetry of the crystal with respect to the defect line is
crucial for proper band engineering of the guided mechanical modes. The two
panels in Fig. 3(b) show the Brillouin spectra measured on the waveguide
(top), and on the surrounding phononic crystal (bottom), as specified in the
insets, taken at the same incident angle of $23.8^{\circ}$. The blue region
indicates the phononic band gap of the structure. The two peaks measured
within this gap in the top panel of Fig. 3(b) at 5.7 GHz and 7.1 GHz are clear
experimental evidence of mechanical vibrations localized in the phononic
waveguide. In order to detect these localized modes, it is necessary to focus
the light on the waveguide with a long-working distance microscope objective
to reduce the spot size of the incident light down to 1.2
$\mathrm{\SIUnitSymbolMicro}$m. In doing so, we reduce the contribution of the
Brillouin scattered signal from the crystal while increasing the contribution
from the waveguide. For the measurements of these waveguide structures, the
background is higher due to a greater collection of reflected and linearly
scattered light relative to that of the 3 cm lens used in Fig. 2. Figure 3(c)
plots the dispersion relation of the waveguide accounting for the $4^{\circ}$
correction of the vertical sidewalls. As in Fig. 2c, the color intensity of
the bands corresponds to the normalized coupling coefficient for the MB
perturbation. The fully shaded regions above and below the gap correspond to
the bulk crystal modes and define the band gap edges of the structure. The
calculated dispersion relation exhibits nine guided modes but only two
(indicated with associated mode profiles) have sufficient mechanical out-of-
plane displacement to potentially be detected in experiment. The horizontal
dashed lines highlight the edges of the band gap measured in Fig. 3(b) and the
black dots correspond to the frequencies of the guided modes measured at
different angles. The angle $\theta=23.8^{\circ}$ corresponds to a normalized
wavevector of 1.34 (replacing $q_{\parallel}=n\pi/a$ in Eq. 1 and solving n),
or $k_{x}/(\pi/a)=0.66$ over the first periodic zone of the waveguide. The two
black dots (1 and 2) coincide with the measured frequencies and wavevector in
Fig. 3(b). We assume that measured peaks around 7 GHz corresponds with the
darker flat mode around 7.7 GHz and not with the lighter curve that it is
spectrally closer to. There is a difference of approximately 700 MHz for this
band while the other peaks agree very closely with the calculated band. The
detection of these two modes is a clear fingerprint of the existence of
localized modes along the Shamrock waveguide.

In summary, we provide direct experimental evidence of the complete absence of
mechanical vibrations at room temperature within a full phononic band gap that
is 5.3 GHz wide with a central frequency of 8.4 GHz, measured using Brillouin
light scattering. This measured mechanical gap is significantly wider than
previous demonstrations in literature, especially at this frequency regime.
Our approach incorporates a geometrical pattern that distributes the mass
within the unit cell, forming mass clusters connected by narrow necks that
result in the destructive interference of phonon waves, giving rise to these
wide mechanical gaps. We achieve control over the width and frequency of the
gap by fine-tuning of the geometrical parameters of the structure, enabling
spectral tunability of the gap from 4 GHz to 12 GHz. This tunability is
extended to the guided modes of a line-defect waveguide, enabling engineering
of the frequency and number of localized modes within the structure. This is
the first demonstration of mechanical guided modes at hypersonic frequencies
in the GHz regime using line-defect waveguides, measured at room temperature
without any external excitation. The structures are fabricated on the standard
SOI platform which enables facile integration into existing photonic systems.
The hypersonic insulator presented here also is a photonic insulator for
Transverse-Electric (TE) modes at telecom wavelengths [24, 25] and can be used
to simultaneously engineer phononic and photonic transport enhancing the
optomechanical coupling between THz photons and GHz phonons. This makes the
crystal an ideal transducer in photonic circuits with potential applications
in high-speed signal processing [37]. Furthermore, this platform can be used
in applications and physical processes in which a wide mechanical band gap is
required to isolate the system from thermal damping, such as in quantum cavity
optomechanics or organic molecular systems [38].

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Acknowledgements

This project has received funding from the European Union’s H2020 FET
Proactive project TOCHA (No. 824140) and Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No. 754558. The
ICN2 authors acknowledge funding by the Severo Ochoa program from Spanish
MINECO (No. SEV-2019-0706), Plan Nacional RTI2018-093921-A-C44 (SMOOTH), and
MCIN project SIP (PGC2018-101743-B-100), as well as by the CERCA Programme
Generalitat de Catalunya. O.F. and G.A. are supported by BIST PhD.
Fellowships, R.C.N. by a Marie Sklodowska-Curie fellowship (No. 897148), and
P.D.G. by a Ramon y Cajal fellowship No. RyC-2015-18124. M.A. and S.S.
gratefully acknowledge funding from the Villum Foundation Young Investigator
Program (No. 13170), the Danish National Research Foundation (No. DNRF147 -
NanoPhoton), Innovation Fund Denmark (No. 0175-00022 - NEXUS), and Independent
Research Fund Denmark (Grant No. 0135-00315 - VAFL).

Author contributions

O.F. designed, simulated, and characterized the samples. M.A. and S.S.
fabricated the samples. G.A., R.C.N., and J.G.B. contributed to the data
analysis. C.M.S.T. and P.D.G. supervised the work. P.D.G. conceived the idea
and the project. O.F. and P.D.G. wrote the manuscript with contributions and
input from all authors.

Competing financial interests

The authors declare no competing financial interests.

Methods

Methods are available in the supplementary information.

Data availability

Data is available upon reasonable request.

## SUPPLEMENTARY INFORMATION

## S1. CRYSTAL DESIGN, FABRICATION AND CHARACTERIZATION

### Shamrock design

The periodic pattern of the phononic crystal consists of three tangential
circles of radius $r=0.22a$, where $a$ is the period of the structure, shifted
by a distance $f=2r/\sqrt{3}$ (Fig. S1) from the common center and arranged to
form what we call a shamrock [1]. Three small semicircles of radius $f-r$ have
been added in the junctions to smooth out any sharp corners in the geometry
and create a structure that can be fabricated.

Figure S1: Shamrock geometry Three adjacent circles of radius $r$ form the
replication pattern of the shamrock phononic crystal.

### Fabrication

All the structures studied here were fabricated in chips cleaved from a
commercial 12-inch silicon-on-insulator (SOI) wafer, which has a nominal 220
nm thick device layer and a 2 $\mathrm{\SIUnitSymbolMicro}$m buried oxide. The
lithography, dry-etching, and release etch of the suspended membrane is
carried out as detailed in [2] with some minor changes. Since we do not need
to make as small features, we spin-coat a thicker (180 nm) softmask
(chemically semi-amplified resist, CSAR6200.09), which enables us to reduce
the periodic sidewall roughness (scallops). We define the patterns with a 100
keV 100 MHz JEOL-9500FSZ electron-beam writer with a current 200 pA and a dose
density of 2.5 aC/nm2 in the center, boosted by 10 % in the corners to
compensate long-range proximity effects across the large crystals [3]. We
consider the variance of the long-range effects, $\beta$ =
30$\mathrm{\SIUnitSymbolMicro}$m, and the relative weight compared to the
forward scattering, $\eta=0.5$. We further modify the cyclic dry-etching
process CORE [4], with 3 changes from the process in Ref. [2]. Specifically,
we increase the number of cycles to 14 and reduce the time of the E-step
(etch) to 45 s. to reduce the size of the scallops. In order to reduce
sidewall erosion, we increase the time of the O-step (passivation) to 5 s. The
structures are released by a 2.8 $\mathrm{\SIUnitSymbolMicro}$m isotropic
underetch of the buried oxide [2].

Figure S2 shows the fabricated samples with periods $a$ of 220, 330 and 440
nanometers with the same radius $r=0.22a$. All the crystals have a full area
of 50x50 $\mathrm{\SIUnitSymbolMicro}m$.

Figure S2: Fabricated samples Scanning-electron micrographs of the shamrock
phononic crystals with period $a$ of (a) 220 nm, (b) 330 nm, and (c) 440 nm.

### Contour fitting and statistical analysis

The average shape of the fabricated shamrock holes is obtained from an
ensemble of 56 holes extracted from the same SEM image. Each contour comes
from a binarized version of the original SEM image adjusted to obtain the most
accurate result as shown Fig. S3(a). Figure S3(b) plots the ensemble of all
contours (red dots) and its mean value (blue dots). The shape of the
individual fitted shamrocks are very similar which is a qualitative measure of
the fabrication tolerance. To quantify the degree of imperfection, we compare
the mean value and standard deviation of the ensemble averaged area of the
shamrocks with the nominal value. Fig. S3(c) plots the histogram distribution
of the areas in nm2 of each individual shamrock (in red), and the mean value
(in blue). The gray line in Fig. S3(c) indicates the nominal area of the
designed geometry. The difference between the average area, $5.12\times
10^{4}\text{nm}^{2}$, and the nominal area, $4.91\times 10^{4}\text{nm}^{2}$,
is attributed to the not fully resolved corners of the shamrock, as displayed
in the inset of the figure. This clearly shows the significance of avoiding
sharp corners in the unit cell geometry, which requires a much accurate
fabrication process. Although geometric inspection using SEM images has its
limitations, this simple analysis shows the high quality of the fabrication
process used here and also the simplicity and robustness of the pattern
selected to create the periodicity.

Figure S3: Fitting and statistical analysis (a) Fitted contours in a shamrock
crystal with a period of 330 nm. (b) Superposition of all the contours (red)
and its mean value (blue points). (c) Histogram for the fitted areas and its
mean, compared with the nominal value of the designed area. There is a small
difference caused by the etching in the intersection of the circles.

## S2. Band structure of the irreducible Brillouin zone

A simple periodic geometry can be created in a material using electron-beam
lithography by etching circularly symmetric holes (C1), often in a hexagonal
lattice (C6v) [5], leading to crystals that belong to the planar space-group
(also called wallpaper group) p6mm. However, different (more restricted)
symmetries of the pattern in the unit cell and the symmetries of the material
at the atomic scale might induce a discrete set of rotational symmetries in
addition to their translation invariance. Figure S4(a-c) illustrates how the
existing symmetries of the unit cell of a triangular lattice give rise to
reduced sizes of the Brillouin zone (BZ) via some examples, where the top
panel represents the real-space Wigner-Seitz cell and the bottom panel the
corresponding first BZ. The circle in Fig. S4(a) possesses full rotational
symmetry (C1) and the 12 operations that map the hexagonal Wigner-Seitz cell
to itself are preserved, leading to an irreducible Brillouin zone (IBZ) (the
shaded region inside $\Gamma KM$) of size 1/12, in units of the first BZ. The
equilateral triangle in S4(b) breaks the inversion symmetry (C2) and three
mirror symmetries ($\widehat{\sigma}_{y}$ , $\widehat{\sigma}_{y^{\prime}}$
and $\widehat{\sigma}_{y^{\prime\prime}}$) leading to an IBZ two times larger.
Other geometries such as a square can preserve inversion symmetry but break a
higher number of mirror symmetries, leading to an IBZ one quarter of the first
BZ in size (c). The shamrock pattern used here [1], preserves the symmetry
described in S4(b). However, if we consider the anisotropy of the silicon
stiffness tensor, the mirror symmetry of the unit cell is broken resulting in
the symmetry described in S4(c). This leads to a IBZ given by the region
bounded by the path $\Gamma KMK_{1}M_{1}\Gamma$. Figure S4(d) plots the
phononic dispersion relation of the system presented in Fig. 1 of the main
text along this IBZ. For simplicity, in our experiment we measure along the
simplified path $\Gamma KM\Gamma$ indicated in Fig. S4(b). This is sufficient
to characterize the full width of the phononic gap as its limiting frequencies
do not change with respect to the IBZ.

Figure S4: Irreducible Brillouin Zone (IBZ). The symmetries of the unit cell
pattern that map the lattice into itself are called the point group of the
crystal and define the IBZ, the minimum region in k-space that needs to be
sampled to access the full eigenspectrum $\omega_{n}(k)$. (a) A circle
preserves all mirror symmetries
$\sigma_{xx^{\prime};x^{\prime\prime};yy^{\prime};y^{\prime\prime}}$, all
rotational symmetries $C_{\pm 6;\pm 3;\pm 2}$ and the inversion symmetry
$C_{\pm 2}$ of the hexagonal Wigner-Seitz (WS) cell of a triangular lattice,
leading to a IBZ delimited by $\Gamma KM$ IBZ, shaded in brown. (b) An
equilateral triangle breaks the $\sigma_{x^{\prime};y;y^{\prime}}$ mirror
symmetries and the inversion symmetry $C_{\pm 2}$, which leads to the shaded
region. Use of time-reversal invariance finally leads to the same $\Gamma KM$
IBZ. Larger IBZs occur when the pattern breaks additional symmetries, as does
(c) a square leading to an IBZ of size 1/4 in units of the 1BZ area. (d)
Calculated mechanical dispersion relation along the the borders of the
irreducible Brillouin zone $\Gamma KMK_{1}M_{1}\Gamma$ for the phononic
crystal presented in Fig. 1a of the main text. Blue and red curves represents
symmetric and asymmetric modes with respect to the mid plane of the slab.

## S3. Brillouin light scattering spectroscopy

The setup used to measure the spontaneous backward Brillouin scattering was a
tandem Fabry-Perot interferometer. A 532 nm continuous-wave solid state laser
with a linewidth of 5 MHz was used for the incident and reference beam. The
scattered light is routed into a six-times pass Fabry-Perot interferometer
that is comprised of two cavities in series, FP1 and FP2 in Fig. S5. This
system increases the spectral resolution and allows high stability and
accurate measurements of spectra over long acquisition times. For measurements
on the crystal, the incident power was 2.5 mW using a 3 cm focal lens, with an
acquisition time of 24 hours for each measured angle. For measurements on the
waveguide and the surrounding shield, the incident power was 1 mW using a long
working distance objective lens with 100x magnification, also with an
acquisition time of 24 hours for each spectrum.

Figure S5: Brillouin light scattering spectroscopy. simplified scheme of the
tandem Fabry-Perot interferometer used to measure and enhance the
backscattered signal. BS: beam spliter, M: mirror, P: prism, FP: Fabry-Perot
cavity.

## S4. SCATTERING EFFICIENCY

The scattering efficiency quantifies how much a mechanical mode scatters the
incident light. To calculate it, we need to consider the origin of the
periodic perturbation of the refractive index. It is caused by two main
physical mechanisms. The most well-known is the photoelastic effect (PE)
induced by the elastic strain of the material [6], and the moving boundary
effect (MB) induced by the mechanical displacement of the surface [7]. The
latter becomes important when light is on the surface, such as in our case.
Here we made a first approximation taking into account only the MB
perturbation and neglecting the PE contribution, given the small volume of
interaction in our experiment. This approximation allows us to obtain the
coupling coefficients in an almost fully analytical way. According to [7], the
MB coupling is given by:

$\kappa_{mb}=\int_{A}U_{z}\left[\Delta\epsilon_{12}\left(\textbf{E}_{s,\parallel}^{*}\cdot\textbf{E}_{i,\parallel}\right)-\Delta\epsilon_{12}^{-1}\left(\textbf{D}_{s,\perp}^{*}\cdot\textbf{D}_{i,\perp}^{{}^{\prime}}\right)\right]dA$
(S1)

where $U_{z}$ is the normal displacement to the surface $A$ as indicated in
Fig. S6(a). $\Delta\epsilon_{12}=(\epsilon_{1}-\epsilon_{2})$ and
$\Delta\epsilon_{12}^{-1}=(\epsilon^{-1}_{1}-\epsilon^{-1}_{2})$, E and D are
electric and displacement fields, and scripts $s$ and $i$ denote scattered and
incident fields respectively. Taking into account the relation between the
relative permittivity and the refractive index, $\epsilon=\epsilon_{0}n^{2}$,
we can write

$\Delta\epsilon_{12}=\epsilon_{0}(n_{1}^{2}-n_{2}^{2})$ (S2)
$\Delta\epsilon_{12}^{-1}=\frac{1}{\epsilon_{0}}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right).$
(S3)

Replacing the normal component of the displacement field by the electric field
component, $\textbf{D}=\epsilon\textbf{E}$, we can rewrite Eq. S1 as

$\kappa_{mb}=\int_{A}U_{z}\epsilon_{0}\left[\left(n_{1}^{2}-n_{2}^{2}\right)\left(\textbf{E}_{s,\parallel}^{*}\cdot\textbf{E}_{i,\parallel}\right)-n_{2}^{4}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)\left(\textbf{E}_{s,\perp}^{*}\cdot\textbf{E}_{i,\perp}^{{}^{\prime}}\right)\right]dA.$
(S4)

The incident and scattered electric field in the sample surface can be treated
as planes waves propagating in free space and can be written as:

$\textbf{E}_{i}(r,t)=\frac{1}{2}\textbf{E}_{i}(x,y)e^{-i(\omega_{i}t-\textbf{k}_{i}\cdot\textbf{z})}+c.c.$
(S5)
$\textbf{E}_{s}(r,t)=\frac{1}{2}\textbf{E}_{s}(x,y)e^{-i(\omega_{s}t+\textbf{k}_{s}\cdot\textbf{z})}+c.c.$
(S6)

We need to account for the polarization of the electric field used in the
experiment (TM polarization) as depicted in Fig. S6. For the backward
Brillouin scattering configuration, $\vec{k}_{s}=-\vec{k}_{i}$ and
$\omega_{s}=\omega_{i}\pm\Omega$ or $\omega_{s}\approx\omega_{i}$ given the
small frequency of mechanical modes (GHz) compared with the incident light
frequency (THz). $\textbf{E}_{s}(x,y)=\textbf{E}_{i}(x,y)=\textbf{E}_{i}$
because light is propagating in free space. Therefore the product of the
incident and scattered fields can be simplified to
$\textbf{E}^{*}_{s}\cdot\textbf{E}_{i}=E^{2}_{i}$. Incident and scattered
light can then be treated as the same fields.

Figure S6: Optomechanical coupling mediated by the moving-boundary effect. (a)
Components of the incident and backscattered electromagnetic field (green and
red respectively). Each mode of the structure leads to a particular vertical
displacement of the surface that determines the scattering contribution. (b)
and (c) highlight the surface to be integrated in the phononic crystal and
phononic waveguide respectively.

As can be seen in Fig. S6(a), we have
$\textbf{E}_{s,\parallel}=-\textbf{E}_{i,\parallel}$ and
$\textbf{E}_{s,\perp}=-\textbf{E}_{i,\perp}$, where the parallel and
perpendicular components of electric field are given by
$\textbf{E}_{i,\parallel}=\textbf{E}_{i}\cos{\theta}$, and
$\textbf{E}_{i,\perp}=\textbf{E}_{i}\sin{\theta}$. Replacing all these
expressions in Eq. S4, we can write

$\kappa_{mb}=\int_{A}U_{z}\epsilon_{0}\left[\left(n_{1}^{2}-n_{2}^{2}\right)\left(-E_{i}^{2}\cos^{2}{\theta}\right)-\left(n_{2}^{2}-n_{1}^{2}\right)\frac{n_{2}^{2}}{n_{1}^{2}}\left(-E_{i}^{2}\sin^{2}{\theta}\right)\right]dA.$
(S7)

The entire expression inside brackets can be taken out the integral. We only
need to account for the change of the refractive index with respect to the
surface displacement. To do so, we always assume a positive change of the
refractive index and take the absolute value of the normal displacement
$\textbf{U}_{z}$.

$\kappa_{mb}=-\epsilon_{0}E^{2}_{i}\left(n^{2}_{1}-n^{2}_{2}\right)\left[\cos^{2}{\theta}+\frac{n^{2}_{2}}{n^{2}_{1}}\sin^{2}{\theta}\right]\int_{A}|U_{z}|dA.$
(S8)

Here the coupling coefficient does not depend on the incident field and it can
be normalized to $E^{2}_{i}=1W$. Each mechanical mode has its own scattering
efficiency depending on the surface displacement $U_{z}$. The bigger the
refractive index contrast, the bigger the MB contribution. Finally, the
scattering cross section needs to be accounted for. The illuminated area on
the sample surface changes with the angle, being a circle of area $A_{c}=\pi
b^{2}$ for $\theta=0$, where b is the beam diameter, and an ellipse with axes
$b$ and $l=b/cos(\theta)$, and area $A_{e}=\pi bl$ for a certain angle
$\theta$. Therefore, Eq. S8 needs to be divided by $cos(\theta)$ to account
for the scattering cross section. Finally, replacing $n_{1}=n_{si}$ and
$n_{2}=1$, the coupling coefficient for the MB perturbation is given by

$\kappa_{mb}=-\epsilon_{0}E^{2}_{i}\left(n_{Si}^{2}-1\right)\left[\cos{\theta}+\frac{1}{n_{Si}^{2}}\tan{\theta}\sin{\theta}\right]\int_{A}|U_{z}|dA.$
(S9)

For the phononic crystal, the area integral has to be computed over the top
face of the unit cell as indicated by Fig. S6(b). For the waveguide, we are
interested in the localized modes; therefore the area integral is not
calculated in the unit entire cell b ut only in the central region as
indicated in Fig. S6(c). Although the expression presented here does not
account for the PE contribution, it is a good approximation to determine if a
particular mechanical mode can be detected in our experimental setup. Equation
S9 is used to calculate the color intensity of bands in Fig. 2(c) and Fig.
3(c) in the main text.

## S5. SPECTRAL TUNABILITY

We explored the geometrical tunability of the gap induced by the variation of
the lattice constant $a$ (see inset of Fig. S7(a)). The spectral evolution of
the gap width as a function of period $a$ in Fig. S7(a) was calculated with a
finite element method assuming a slab thickness of 220 nm and a unit cell fill
fraction of $r/a$ = 0.22. Here we do not account for or apply any correction
to the vertical profile of the crystal. The gap opens for a lattice unit of
$a$ = 100 nm and has a maximum of almost 6 GHz for $a$ = 300 nm. We confirm
the mechanical spectral shift by measuring the Brillouin inelastic scattering
spectrum at the high-symmetry point $K$ for three different structures
fabricated on silicon-on-insulator with different periods of $a$ = 220 nm, 330
nm and 440 nm, as shown Fig. S7(b). We measure at angles which correspond to
$K$ at $\theta=53.7^{\circ}$, $32.5^{\circ}$, and $23.8^{\circ}$,
respectively. The agreement between simulations and experiment in Fig. S7 is
strong evidence of the gap displacement. As mentioned previously, differences
in frequency between theory and experiment are due to unavoidable fluctuations
in the fabrication process which become more significant for smaller lattice
units. The broad tunability of the mechanical gaps measured here, from 4 GHz
to 12 GHz, enables a fine control of hypersonic phonon routing and cavity
optomechanics based on the shamrock geometry.

Figure S7:  Geometric tunability of the system (a) Spectral evolution of the
mechanical gap as a function of the lattice period $a$, considering the same
unit cell fill fraction, $r/a=0.22$. (b) Brillouin light scattering spectra
measured for three different crystals with periods $a$ = 220 nm, 330 nm, and
440 nm. The light-blue regions highlight the spectral position of the
mechanical gap in each structure.

## References

  * [1] I. Söllner, L. Midolo, and P. Lodahl. Deterministic Single-Phonon Source Triggered by a Single Photon. Phys. Rev. Lett. 116, 234301 (2016).
  * [2] M. Albrechtsen et al. Nanometer-scale photon confinement inside dielectrics. See preprint at arXiv:2108.01681 (2021).
  * [3] G. Owen and P. Rissman. Proximity effect correction for electron beam lithography by equalization of background dose. J. Appl. Phys. 54, 3573 (1983).
  * [4] V. T. H. Nguyen et al. The CORE sequence: a nanoscale fluorocarbon-free silicon plasma etch process based on SF6/O2 cycles with excellent 3D profile control at room temperature. ECS J. Solid State Sci. Technol. 9, 024002 (2020).
  * [5] K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed., edited by W. Rhodes (Springer Berlin Heidelberg New York, 2005) p. 253.
  * [6] D. F. Nelson, and M. Lax. Theory of the photoelastic interaction. Phys. Rev. B. 3, 2778 (1971).
  * [7] S. G. Johnson et al. Perturbation theory for Maxwell’s equations with shifting material boundaries. Phys. Rev E. 65, 066611 (2002).


# Maximal intersections in finite groups

Andrea Lucchini  Università degli Studi di Padova
Dipartimento di Matematica “Tullio Levi-Civita”
Via Trieste 63, 35121 Padova, Italy
email<EMAIL_ADDRESS>
###### Abstract.

For a finite group $G$, we investigate the behaviour of four invariants,
$\text{MaxDim}(G),$ $\text{MinDim}(G),$ $\text{MaxInt}(G)$ and
$\text{MinInt}(G),$ measuring in some way the width and the height of the
lattice $\mathcal{M}(G)$ consisting of the intersections of the maximal
subgroups of $G.$

###### Key words and phrases:

finite groups, subgroup lattice, maximal subgroups

###### 1991 Mathematics Subject Classification:

20D30, 20D10, 20E28

We would like to thank Silvio Dolfi, Daniele Garzoni and Attila Maróti for
fruitful discussions and valuable and helpful comments.

We will say that a subgroup $H$ of a finite group $G$ is a maximal
intersection in $G$ if there exists a family $M_{1},\dots,M_{t}$ of maximal
subgroups of $G$ with $H=M_{1}\cap\dots\cap M_{t}.$ We will denote by
$\mathcal{M}(G)$ the subposet of the subgroup lattice of $G$ consisting of $G$
and all the maximal intersections in $G$.

Let $\mathcal{X}$ be a set of maximal subgroups of the finite group $G.$ We
say that $\mathcal{X}$ is irredundant if the intersection of the subgroups in
$\mathcal{X}$ is not equal to the intersection of any proper subset of
$\mathcal{X}.$ The maximal dimension $\operatorname{MaxDim}(G)$ of $G$ is
defined as the maximal size of an irredundant set of maximal subgroups of $G.$
This definition arises from the study of the maximum size $m(G)$ of an
irredundant generating set for $G$ (that is, a generating set that does not
properly contain any other generating set). Indeed, it is easy to see that
$m(G)\leq\operatorname{MaxDim}(G)$. However in [6] it was proved that the
difference $\operatorname{MaxDim}(G)-m(G)$ can be arbitrarily large.
Independently on this motivation, $\operatorname{MaxDim}(G)$ can be view as a
measure of the width of the poset $\mathcal{M}(G).$ The study of the maximal
dimension of an arbitrary finite group is quite complicated and it is
difficult to find good strategies to investigate this invariant. For example
one could ask the following question.

######  Question 1.

Let $N$ be a normal subgroup of a finite group $G.$ Suppose that
$\delta=\operatorname{MaxDim}{G/N},$ $d=\operatorname{MaxDim}{G}$ and
$\\{M_{1}/N,\dots,$ $M_{\delta}/N\\}$ is an irredundant set of maximal
subgroups of $G/N$. Do there exist $d-\delta$ maximal subgroups
$M_{\delta+1},\dots,M_{d}$ of $G$ such that $M_{1},\dots,M_{\delta},$
$M_{\delta+1},\dots,M_{d}$ is an irredundant set of maximal subgroups of $G$?

We will prove that the answer is negative and this makes difficult to estimate
$\operatorname{MaxDim}(G)$ arguing by induction. Another natural question to
which we will give an unexpected negative answer is the following.

######  Question 2.

Does $G$ contain an irredundant family of maximal subgroups of size
$\operatorname{MaxDim}(G)$ whose intersection is the Frattini subgroup?

The dual concept of minimal dimension was introduced in [8]. We say that an
irredundant set of maximal subgroups is maximal irredundant if it is not
properly contained in any other irredundant set of maximal subgroups. Then the
minimal dimension of $G,$ denoted $\operatorname{MinDim}(G)$, is the minimal
size of a maximal irredundant set. In [3] it was proposed to study the finite
groups $G$ with $\operatorname{MinDim}(G)=\operatorname{MaxDim}(G)$ (minmax
groups). All nilpotent groups are minmax, but there are non-nilpotent minmax
groups, such as Sym(3), Alt(4) and Sym(4). By a well-known theorem of Iwasawa
[12], all unrefinable chains in the subgroup lattice of a finite group $G$
have the same length if and only if $G$ is supersoluble. In our case, in place
of arbitrary unrefinable chains, we restrict our attention to the unrefinable
chains in the poset $\mathcal{M}(G).$ In the context of Iwasawa’s result, it
is worth noting that supersolubility does not imply the minmax property.
However it seems reasonable to conjecture that every minmax group is soluble.
The results in [3] give evidence to this conjecture, proving that every non-
abelian finite simple group is not minmax.

One could even expect to have a relation between $\operatorname{MinDim}(G),$
$\operatorname{MaxDim}(G)$ and the minimal and maximal length of an
unrefinable chain in $\mathcal{M}(G)$. We call these two new invariants
$\operatorname{MinInt}(G)$ and $\operatorname{MaxInt}(G)$. However the
behavior of these invariants is more intricate than one can expected. It is
not difficult to prove that the following holds.

######  Theorem 3.

If $G$ is a finite group, then

  1. (1)

$\operatorname{MinDim}(G)\leq\operatorname{MinInt}(G);$

  2. (2)

$\operatorname{MaxDim}(G)\leq\operatorname{MaxInt}(G).$

However the differences $\operatorname{MinInt}(G)-\operatorname{MinDim}(G)$
and $\operatorname{MaxInt}(G)-\operatorname{MaxDim}(G)$ can be arbitrarily
large. We will see in Section 2 that for any pair $(a,b)$ of positive integers
with $2\leq a\leq b,$ it can be constructed a finite soluble group $G$ with
$\operatorname{MinDim}(G)=a+b,$ $\operatorname{MinInt}(G)=2a+b,$
$\operatorname{MaxDim}(G)=a+b,$ $\operatorname{MaxInt}(G)=a+2b.$

It is more difficult to compare $\operatorname{MaxDim}(G)$ and
$\operatorname{MinInt}(G).$ In the example mentioned above
$\operatorname{MinInt}(G)-\operatorname{MaxDim}(G)=a$ can be chosen to be
arbitrarily large. However if $p$ is a prime, then
$\lim_{p\to\infty}\operatorname{MaxDim}(\operatorname{Sym}(p))-\operatorname{MinInt}(\operatorname{Sym}(p))=\infty.$

######  Definition 4.

We say that a finite group $G$ is strongly minmax if
$\operatorname{MaxDim}(G)=\operatorname{MinInt}(G)$.

By Theorem 3, if $G$ is strongly minmax then $\operatorname{MinDim}(G)=$
$\operatorname{MaxDim}(G)$
$=\operatorname{MinInt}(G)=\operatorname{MaxInt}(G).$ This occurs for example
when $G$ is nilpotent or, more in general, if there exists a finite nilpotent
group $K$ with $\mathcal{M}(G)\cong\mathcal{M}(K).$ The group with these
property have been studied in [11], where it is proved in particular that they
are supersoluble. Notice that $\operatorname{Sym}(4)$ is a strongly minmax
group which is not supersoluble.

######  Definition 5.

Let $G$ be a finite group. We define $\alpha(G)$ as the smallest cardinality
of a family of maximal subgroups of $G$ with the property that their
intersection coincide with the Frattini subgroup of $G$.

Clearly $\operatorname{MinDim}(G)\leq\alpha(G)\leq\operatorname{MaxInt}(G).$
In particular, if $G$ strongly minmax, then
$\operatorname{MinDim}(G)=\alpha(G)=\operatorname{MaxInt}(G).$ This motivates
the following definition.

######  Definition 6.

We say that a finite group $G$ is weakly minmax if
$\operatorname{MinInt}(G)=\operatorname{MaxInt}(G)=\alpha(G).$

Our main theorem is the following.

######  Theorem 7.

If $G$ if a finite weakly minmax group, then $G$ is soluble. Moreover the
derived length of $G/\operatorname{Frat}(G)$ is at most 3.

The bound 3 on the derived length is best possible, since, for example,
$\operatorname{Sym}(4)$ is weakly minmax. Notice that
$\operatorname{MinInt}(\operatorname{Alt}(5))=\operatorname{MaxInt}(\operatorname{Alt}(5))=3,$
hence the condition $\operatorname{MinInt}(G)=\operatorname{MaxInt}(G)$ does
not imply that $G$ is soluble.

##  1\. Negative answers to questions 1 and 2

Our first aim is to give a negative answer to question 1. Let $\mathbb{F}$ be
the field with $11$ elements and let $C=\langle c\rangle$ be the subgroup of
order $5$ of the multiplicative group of $\mathbb{F}.$ Let $V=\mathbb{F}^{5}$
be a $5$-dimensional vector space over $\mathbb{F}$ and let
$\sigma=(1,2,3,4,5)\in\operatorname{Sym}(5).$ The wreath group
$H=C\wr\langle\sigma\rangle$ has an irreducible action on $V$ defined as
follows: if $v=(f_{1},\dots,f_{5})\in V$ and $h=(c_{1},\dots,c_{5})\sigma\in
H$, then
$v^{h}=(f_{1\sigma^{-1}}c_{1\sigma^{-1}},\dots,f_{5\sigma^{-1}}c_{5\sigma^{-1}}).$
We will concentrate our attention on the semidirect product $G:=V\rtimes H$
(notice that $G=G_{11,5}$ in the notations of [6, Section 3]). By [6,
Proposition 11] $\operatorname{MaxDim}(G)=5$, while $H\cong G/V$ is a
2-generated nilpotent group with $\operatorname{MaxDim}(H)$=2. Let
$M_{1},M_{2}$ be two different maximal subgroups of $G$ containing $V.$ We
have $M_{1}=V\rtimes K_{1}$ and $M_{2}=V\rtimes K_{2}$ with $K_{1}$ and
$K_{2}$ maximal subgroups of $H.$ Assume, by contradiction, that
$\\{M_{1},M_{2}\\}$ can be lifted to an irredundant set
$\\{M_{1},M_{2},M_{3},M_{4},M_{5}\\}$ of maximal subgroups of $G.$ Then
$M_{3},M_{4},M_{5}$ are complements of $V$ in $G$ and it is not restrictive to
assume $M_{3}=H,$ $M_{4}=H^{v_{1}},$ $M_{5}=H^{v_{2}}$ with
$v_{1}=(x_{1},x_{2},x_{3},x_{4},x_{5}),v_{2}=(y_{1},y_{2},y_{3},y_{4},y_{5})\in
V.$ We must have $|M_{3}\cap M_{4}|\geq 5^{3},$ hence, by [6, Lemma 10], the
subset $I$ of $\\{1,\dots,5\\}$ consisting of the indices $i$ with $x_{i}=0$
contains at least 3 elements and $M_{3}\cap
M_{4}=\\{(c_{1},c_{2},c_{3},c_{4},c_{5})\in C_{5}^{5}\mid c_{i}=1\text{ if
}i\neq I\\}.$ Notice that $M_{1}\cap M_{2}=V\rtimes F$ with
$F=\operatorname{Frat}H=\\{(c_{1},c_{2},c_{3},c_{4},c_{5})\in C_{5}^{5}\mid
c_{1}c_{2}c_{3}c_{4}c_{5}=1\\}.$ In particular $M_{1}\cap M_{2}\cap M_{3}\cap
M_{4}=\\{(c_{1},c_{2},c_{3},c_{4},c_{5})\in C_{5}^{5}\mid
c_{1}c_{2}c_{3}c_{4}c_{5}=1\text{ and }c_{i}=1\text{ if }i\neq I\\},$ but then
$|M_{3}\cap M_{4}:M_{1}\cap M_{2}\cap M_{3}\cap M_{4}|=5$, a contradiction.

Now we give a negative answer to question 2. Let $G=\operatorname{AGL}(2,5),$
$N=\operatorname{soc}(G)$ and $F/N=\operatorname{Frat}(G/N).$ We have $N\cong
C_{5}\times C_{5}$, $F/N\cong\operatorname{Frat}(\operatorname{GL}(2,5))\cong
C_{4}$ and $G/F\cong\operatorname{Sym}(5).$ For a maximal subgroup $M$ of $G$
we have the following possibilities:

  1. (1)

$M$ is a complement of $N$ in $G$ (25 conjugates);

  2. (2)

$F\leq M$ and $M/F\cong\operatorname{Alt}(5)$ (1 conjugate);

  3. (3)

$F\leq M$ and $M/F\cong\operatorname{Sym}(4)$ (5 conjugates);

  4. (4)

$F\leq M$ and $M/F\cong C_{5}\rtimes C_{4}$ (6 conjugates);

  5. (5)

$F\leq M$ and $M/F\cong\operatorname{Sym}(3)\times\operatorname{Sym}(2)$ (10
conjugates).

Let $H$ be a complement of $N$ in $G$ and let $\\{K_{1},\dots,K_{4}\\}$ be an
irredundant family of maximal subgroups of $G$ of type 3. Then $\\{H\cap
K_{1},\dots,H\cap K_{4}\\}$ is an irredundant family of maximal subgroups of
$H$ and $H\cap K_{1}\cap\dots\cap K_{4}=H\cap F=\operatorname{Frat}(H)\cong
C_{4}.$ In particular $\operatorname{MaxDim}(G)\geq 5.$ Now assume that
$\mathcal{M}=\\{M_{1},\dots,M_{t}\\}$ is an irredundant family of maximal
subgroups of $G$ with $M_{1}\cap\dots\cap M_{t}=1.$ At least one of these
maximal subgroups of $G$ must be of type (1), otherwise $M_{1}\cap\dots\cap
M_{t}\geq F.$ So it is not restrictive to assume $M_{1}=H.$ We distinguish two
possibilities:

a) $M_{i}$ is not of type (1), whenever $i\geq 2.$ In this case, for $i\geq 2$
there exists a maximal subgroup $K_{i}$ of $H$ such that $M_{i}=NK_{i}$. But
then $M_{1}\cap M_{2}\cap\dots\cap M_{t}=K_{2}\cap\dots\cap
K_{t}\geq\operatorname{Frat}(H)\cong C_{4},$ a contradiction.

b) $M_{2}$ is of type (1). We have $M_{2}=H^{n}$ for some $1\neq n\in N$ and
$D=M_{1}\cap M_{2}=H\cap H^{n}=C_{H}(n)\cong C_{5}\rtimes C_{4}.$ If $X$ is a
maximal subgroup of $G$ of type (3), then $X\cap D\cong C_{4}$, and
consequently, if $X_{1}$ and $X_{2}$ are two different maximal subgroups of
type (3), then $X_{1}\cap X_{2}\cap D=1$. Hence either $t\leq 4$ or
$\mathcal{M}$ contains at most one maximal subgroup of type (3). One of the 6
maximal subgroups of $G$ of type (4) contains $D,$ the other intersect $D$ in
a subgroup of order $4$, moreover if $Y_{1}$ and $Y_{2}$ are two different
maximal subgroups of type (4) not containing $D$, then $Y_{1}\cap Y_{2}\cap
D=1$. Hence either $t\leq 4$ or $\mathcal{M}$ contains at most one maximal
subgroup of type (4). If $Z$ is a maximal subgroup of $G$ of type (5), then
$Z\cap D\cong C_{2}$. Hence if $\mathcal{M}$ contains a maximal subgroup of
type (5) then $t\leq 4.$ Summarizing we have proved that either $t\leq
4<\operatorname{MaxDim}(G)$ or $t=5$ and in that case we may assume $M_{3}$ of
type (2), $M_{4}$ of type (3) and $M_{5}$ of type (4). However this case
cannot occur since it can be easily checked that if $X$ is of type (3) and $Y$
is of type (4) and does not contains $D$, then either $X\cap Y\cap D=1$ or
$X\cap D=Y\cap D.$

Although question 2 has a negative answer, a weaker result holds.

######  Proposition 8.

If $\\{M_{1},\dots,M_{t}\\}$ is a maximal irredundant family of maximal
subgroups, then $\operatorname{Core}_{G}(M_{1}\cap\dots\cap
M_{t})=\operatorname{Frat}(G).$

###### Proof.

It is not restrictive to assume $\operatorname{Frat}(G)=1.$ Let
$\\{M_{1},\dots,M_{t}\\}$ be a maximal irredundant family of maximal subgroups
of $G$ and let $X=M_{1}\cap\dots\cap M_{t}.$ Assume, by contradiction, that
$X$ contains a nontrivial normal subgroup, say $N$, of $G.$ Since
$\operatorname{Frat}(G)=1,$ there exists a maximal subgroup $Y$ of $G$ with
$N\not\leq Y.$ Since $\\{M_{1},\dots,M_{t}\\}$ is a maximal irredundant
family, the family $\\{M_{1},\dots,M_{t},Y\\}$ is not irredundant. On the
other hand, since $N\leq M_{1}\cap\dots\cap M_{t},$ we cannot have
$M_{1}\cap\dots\cap M_{t}\leq Y.$ So, up to reordering, we may assume
$M_{1}\cap\dots\cap M_{t-1}\cap Y\leq M_{t}.$ Let $U=M_{1}\cap\dots\cap
M_{t-1}.$ Since $N\leq U,$ by the Dedekind law, $N(U\cap Y)=U\cap NY=U.$ From
$U\cap Y\leq M_{t},$ it follows $U=N(U\cap Y)\leq NM_{t}=M_{t}$, against the
assumption that $\\{M_{1},\dots,M_{t}\\}$ is an irredundant family. ∎

######  Question 9.

Is the answer to question 1 negative also in the case of finite soluble
groups?

##  2\. An Example

Let $H$ be a cyclic group of order $m:=p^{a}q^{b},$ where $p$ and $q$ two
different primes and $2\leq a\leq b.$ For any divisor $r$ of $m,$ denote by
$H_{r}$ the unique subgroup of $H$ of order $r.$ Let
$I:=\\{p,p^{2},\dots,p^{a-1},q,q^{2},\dots,q^{b-1}\\}.$ For any $i\in I,$ let
$p_{i}$ be a prime such that $m/i$ divides $p_{i}-1$ and let $X_{i}\cong
C_{p_{i}}.$ We have an action of $H$ on $X_{i}$ with kernel $H_{i}$. Let
$X:=\prod_{i\in I}X_{i}$ and $G:=X\rtimes H.$ The maximal subgroups of $G$ are
the following:

  1. (1)

$A_{p}:=X\rtimes H_{m/p},$ $A_{q}:=X\rtimes H_{m/q};$

  2. (2)

$B_{p^{r},x}:=(\prod_{i\neq p^{r}}X_{i})\rtimes H^{x}$, with
$r\in\\{1,\dots,a-1\\}$ and $x\in X_{p^{r}};$

  3. (3)

$B_{q^{s},x}:=(\prod_{i\neq q^{s}}X_{i})\rtimes H^{y}$, with
$s\in\\{1,\dots,b-1\\}$ and $y\in X_{q^{s}}.$

Now we study the intersections of these maximal subgroups. Notice that if
$t\geq 2$ and $x_{1},\dots,x_{t}$ are distinct elements of $X_{p^{r}},$ then

(2.1) $B_{p^{r},x_{1}}\cap\dots\cap B_{p^{r},x_{t}}=B_{p^{r},x_{1}}\cap
B_{p^{r},x_{2}}=\left(\prod_{i\neq p^{r}}X_{i}\right)\rtimes H_{p^{r}}.$

Similarly, if $t\geq 2$ and $y_{1},\dots,x_{t}$ are distinct elements of
$X_{q^{s}},$ then

(2.2) $B_{q^{s},x_{1}}\cap\dots\cap B_{q^{s},x_{t}}=B_{q^{s},x_{1}}\cap
B_{q^{s},x_{2}}=\left(\prod_{i\neq q^{s}}X_{i}\right)\rtimes H_{q^{s}}.$

Let $\mathcal{Y}$ be a family of maximal subgroups of $G$. For any $i\in I,$
let $\mathcal{Y}_{i}$ be the set of subgroups in $\mathcal{Y}$ of the form
$B_{i,x},$ for some $x\in X_{i}.$ Moreover define

  * •

$I_{0}(\mathcal{Y}):=\\{i\in I\mid\mathcal{Y}_{i}=\varnothing\\},$

  * •

$I_{1}(\mathcal{Y}):=\\{i\in I\mid|\mathcal{Y}_{i}|=1\\},$

  * •

$I_{2}(\mathcal{Y}):=\\{i\in I\mid|\mathcal{Y}_{i}|>1\\}.$

For any $i\in I_{1}(\mathcal{Y}),$ there exists a unique $x_{i,\mathcal{Y}}\in
X_{i}$ such that $B_{i,x_{i,\mathcal{Y}}}\in\mathcal{Y}.$ Let

$x_{\mathcal{Y}}:=\prod_{i\in I_{1}(\mathcal{Y})}x_{i,\mathcal{Y}}.$

Finally set

$\tau_{p}(\mathcal{Y})=\min\\{a,r\mid p^{r}\in
I_{2}(\mathcal{Y})\\},\tau_{q}(\mathcal{Y})=\min\\{b,s\mid q^{s}\in
I_{2}(\mathcal{Y})\\}$

and define

$L_{\mathcal{Y}}:=\begin{cases}H&\text{if $A_{p},B_{p}\notin\mathcal{Y}$,}\\\
H_{p^{a-1}q^{b}}&\text{if $A_{p}\in\mathcal{Y}$, $B_{p}\notin\mathcal{Y},$}\\\
H_{p^{a}q^{b-1}}&\text{if $A_{p}\notin\mathcal{Y}$, $B_{p}\in\mathcal{Y},$}\\\
H_{p^{a-1}q^{b-1}}&\text{if $A_{p}\in\mathcal{Y}$, $B_{p}\in\mathcal{Y}.$}\\\
\end{cases}$

  1. (1)

If $\tau_{p}(\mathcal{Y})=a$ and $\tau_{q}(\mathcal{Y})=b$, then

$\bigcap_{Y\in\mathcal{Y}}Y=\left(\prod_{i\in
I_{0}(\mathcal{Y})}X_{i}\right)L_{\mathcal{Y}}^{x_{\mathcal{Y}}}.$

In this case
$|\mathcal{Y}|=|I_{1}(\mathcal{Y})|+|\mathcal{Y}\cap\\{A_{p},A_{q}\\}|\leq
a+b+2.$ Moreover, if $\mathcal{Y}$ is a maximal irredundant family, then by
Proposition 8, $I_{0}(\mathcal{Y})=\varnothing$, $I_{1}(\mathcal{Y})=I$ and
$A_{p},A_{q}\in\mathcal{Y}.$ This implies $|\mathcal{Y}|=a+b.$

  2. (2)

If $\tau_{p}(\mathcal{Y})<a$ and $\tau_{q}(\mathcal{Y})=b$, then, setting
$t=\tau_{p}(\mathcal{Y}),$ we have

$\bigcap_{Y\in\mathcal{Y}}Y=\left(\prod_{i\in
I_{0}(\mathcal{Y})}X_{i}\right)H_{p^{t}}^{x_{\mathcal{Y}}}.$

Let $y_{1},y_{2}$ be two different elements in $X_{q}.$ If follows from the
fact that $\cap_{Y\in\mathcal{Y}}Y\cap X_{q,y_{1}}\cap X_{q,y_{2}}=1$ that
$\mathcal{Y}$ cannot be a maximal irredundant family.

  3. (3)

If $\tau_{p}(\mathcal{Y})=a$ and $\tau_{q}(\mathcal{Y})<b$, then, setting
$t=\tau_{q}(\mathcal{Y}),$ we have

$\bigcap_{Y\in\mathcal{Y}}Y=\left(\prod_{i\in
I_{0}(\mathcal{Y})}X_{i}\right)H_{q^{t}}^{x_{\mathcal{Y}}}.$

Let $y_{1},y_{2}$ be two different elements in $X_{p}.$ If follows from the
fact that $\cap_{Y\in\mathcal{Y}}Y\cap X_{p,y_{1}}\cap X_{p,y_{2}}=1$ that
$\mathcal{Y}$ cannot be a maximal irredundant family.

  4. (4)

If $\tau_{p}(\mathcal{Y})<a$ and $\tau_{q}(\mathcal{Y})<b$, then

$\bigcap_{Y\in\mathcal{Y}}\\!Y\\!=\prod_{i\in I_{0}(\mathcal{Y})}\\!\\!X_{i}.$

Moreover, if $\mathcal{Y}$ is a maximal irredundant family, then by
Proposition 8, $I_{0}(\mathcal{Y})=\varnothing$,
$I_{1}(\mathcal{Y})=I\setminus\\{p^{t},q^{u}\\},$
$A_{p},A_{q}\notin\mathcal{Y}.$ This implies $|\mathcal{Y}|=a+b.$

From the previous discussion, it follows:

######  Proposition 10.

$\operatorname{MinDim}(G)=\operatorname{MaxDim}(G)=a+b.$

Clearly we have an unrefinable chain in $\mathcal{M}(G)$ of length $|I|=a+b-2$
from $H$ to $G$ and a chain of length 2 from $H_{p^{a-1}q^{b-1}}$ to $H$.
Moreover we have the following two unrefinable chains from $1$ to
$H_{p^{a-1}q^{b-1}}:$

$1<H_{p}\leq\dots\leq H_{p^{a-1}}\leq H_{p^{a-1}q^{b-1}},\quad\quad
1<H_{q}\leq\dots\leq H_{q^{b-1}}\leq H_{p^{a-1}q^{b-1}}.$

In particular we may easily conclude:

######  Proposition 11.

$\operatorname{MinInt}(G)=2a+b,\quad\operatorname{MaxInt}(G)=a+2b.$

##  3\. Proof of Theorem 3 and further considerations

######  Lemma 12.

Assume that $\\{M_{1},\dots,M_{n}\\}$ is a family of maximal subgroups of $G.$
There exists $J\subseteq I:=\\{1,\dots,n\\},$ such that

  1. (1)

$\\{M_{j}\mid j\in J\\}$ is an irredundant family;

  2. (2)

$\cap_{i\in I}M_{j}=\cap_{j\in J}M_{j}.$

###### Proof.

By induction on $n.$ If $\\{M_{i}\mid i\in I\\}$ is a redundant family, then
there exists $k\in I$ such that $\cap_{i\in I}M_{i}=\cap_{j\in
I\setminus\\{k\\}}M_{j}.$ We substitute the original family with the subfamily
$\\{M_{j}\mid j\in I\setminus\\{k\\}\\}$ and we conclude by induction. ∎

######  Proposition 13.

$\operatorname{MinDim}(G)\leq\operatorname{MinInt}(G).$

###### Proof.

We may assume $\operatorname{Frat}(G)=1.$ Let $t=\operatorname{MinInt}(G)$ and
assume that

$\mathcal{C}:K_{t}<K_{t-1}<\dots<K_{1}<K_{0}=G$

is a non refinable chain in $\mathcal{M}(G).$ There exists $t$ maximal
subgroups $M_{1},\dots,M_{t}$ of $G,$ such that $K_{i}=\cap_{j\leq i}M_{j}$
for $1\leq i\leq t.$ Since $\operatorname{Frat}(G)=1,$ if $K_{t}\neq 1,$ then
there exists a maximal subgroup $M$ not containing $K_{t},$ and consequently
$K_{t}\cap M<K_{t}<\dots<K_{0}$ is a refinement of $\mathcal{C}.$ Hence
$K_{t}=1.$ It follows from Lemma 12 that there exists
$J\subseteq\\{1,\dots,t\\}$ such that $\\{M_{j}\mid j\in J\\}$ is an
irredundant family of maximal subgroups of $G$ with $\cap_{j\in J}M_{j}=1.$
The second condition implies that $\\{M_{j}\mid j\in J\\}$ is a maximal
irredundant family of maximal subgroups of $G,$ hence
$\operatorname{MinDim}(G)\leq t.$ ∎

######  Proposition 14.

$\operatorname{MaxDim}(G)\leq\operatorname{MaxInt}(G).$

###### Proof.

Let $t=\operatorname{MaxDim}(G)$ and suppose that $\\{M_{1},\dots,M_{t}\\}$ is
an irredundant family of maximal subgroups of $G$. For $1\leq j\leq t,$ set
$K_{j}=\cap_{i\leq j}M_{i}:$

$K_{t}<K_{t-1}<\dots<K_{1}<K_{0}$

is a chain in $\mathcal{M}$ of length $t,$ and this implies
$\operatorname{MaxInt}(G)\geq t.$ ∎

It is more difficult to compare $\operatorname{MaxDim}(G)$ and
$\operatorname{MinInt}(G).$ In the example discussed in the Section 2,
$\operatorname{MinInt}(G)-\operatorname{MaxDim}(G)=a.$ On the other hand, if
$G=\operatorname{Sym}(p)$ with $p$ a prime, we may consider the chain
$1<\operatorname{GL}(1,p)<\operatorname{AGL}(1,p)<\operatorname{Sym}(p).$
Since $\operatorname{GL}(1,p)$ is maximal in $\operatorname{AGL}(1,p)$ and
$\operatorname{AGL}(1,p)$ is maximal in $\operatorname{Sym}(p),$ we may refine
this chain to a chain of maximal intersections of length at most
$2+\log_{2}(p-1).$ Since
$\operatorname{MaxDim}(\operatorname{Sym}(p))=m(G)=p-1,$ we have examples of
finite groups $G$ for which the difference
$\operatorname{MaxDim}(G)-\operatorname{MinInt}(G)$ is arbitrarily large. We
may also construct finite soluble groups $G$ with
$\operatorname{MaxDim}(G)>\operatorname{MinInt}(G).$ Indeed assume that $p,q$
an $r$ are three primes and that $p$ divides $r-1.$ Let $\mathbb{F}$ be the
field with $r$ elements and let $C=\langle c\rangle$ be the subgroup of order
$p$ of the multiplicative group of $\mathbb{F}.$ Let $V=\mathbb{F}^{q}$ be a
$p$-dimensional vector space over $\mathbb{F}$ and let
$\sigma=(1,2,\dots,q)\in\operatorname{Sym}(q).$ The wreath group
$H=C\wr\langle\sigma\rangle$ has an irreducible action on $V$ defined as
follows: if $v=(f_{1},\dots,f_{p})\in V$ and $h=(c_{1},\dots,c_{p})\sigma\in
H$, then
$v^{h}=(f_{1\sigma^{-1}}c_{1\sigma^{-1}},\dots,f_{q\sigma^{-1}}c_{q\sigma^{-1}}).$
We consider the semidirect product $G_{q,p,r}=V\rtimes H.$ Let

$e_{1}=(1,0,\dots,0),e_{2}=(0,1,\dots,0),\dots,e_{q}=(0,0,\dots,1)\in V,$
$h_{1}=(c,1,\dots,1),h_{2}=(1,c,\dots,1),\dots,h_{q}=(1,1,\dots,c)\in
C^{q}\leq H.$

For any $1\leq i,j\leq q,$ we have

$h_{i}^{e_{j}}=h_{i}\text{ if }i\neq j,\quad
h_{i}^{e_{i}}=((1/c-1)e_{i})h_{i}.$

But then, for each $i\in\\{1,\dots,q\\},$ we have

$h_{i}\in\bigcap_{j\neq i}H^{e_{j}},\quad h_{i}\notin H^{e_{i}},$

hence $H^{e_{1}},\dots,H^{e_{q}}$ is an irredundant family of maximal
subgroups of $G_{q,p,r}$ and therefore $\operatorname{MaxDim}(G_{p,q,r})\geq
q.$ Now assume that $p$ has order $q-1$ mod $q:$ in that case $C^{q}$ is the
direct sum of the irreducible $\langle\sigma\rangle$-module, $C_{1}$ and
$C_{2}$, of dimension, respectively, $1$ and $q-1.$ If we consider

$Y_{0}=1<Y_{1}=\langle\sigma\rangle<Y_{2}=C_{1}\langle\sigma\rangle<Y_{3}=H<Y_{4}=G_{p,r,q},$

we have that $Y_{i}$ is a maximal subgroup of $Y_{i+1},$ so
$\operatorname{MinInt}(G)\leq 4.$ Hence the difference
$\operatorname{MaxDim}(G)-\operatorname{MinInt}(G)$ can be arbitrarily large
even in the soluble case.

The fact that $m(G)\leq\operatorname{MaxDim}(G)$ motivates the following
question.

######  Question 15.

Does there exist a finite soluble group $G$ with the property that
$m(G)>\operatorname{MinInt}(G)?$

We are going to prove that the previous question has an affirmative answer if
the Fitting length of $G$ is at most 2. But the question remains open in the
general case.

######  Lemma 16.

Let $G$ be a finite nilpotent group and let $\mathcal{F}$ be a family of
subgroups of $G$ which contains $G$ and all the maximal subgroups of $G$ and
is closed under taking intersections. Let $\mathcal{C}=X_{t}<X_{t-1}<\dots
X_{1}<X_{0}$ be a chain in $\mathcal{F}$ . If $\mathcal{C}$ cannot be refined
in $\mathcal{F}$, then $t\geq u,$ where $u$ is the composition length of
$G/\operatorname{Frat}(G).$

###### Proof.

Clearly, since $\mathcal{C}$ is not refinable, we must have $X_{0}=G.$ Let
$F=\operatorname{Frat}(G).$ If $X_{t}\not\leq F,$ then there exists a maximal
subgroup $M$ of $G$ not containing $X_{t}$. But then $X_{t}\cap M<X_{t}$.
However $X_{t}\cap M\in\mathcal{F},$ and this would contradict the assumption
that $\mathcal{C}$ cannot be refined in $\mathcal{F}.$ For any $H\leq G,$ let
$\overline{H}:=HF/F.$ We have a chain

$\overline{\mathcal{C}}:\ F=\overline{X}_{t}\leq\dots\leq\overline{X}_{0}.$

Assume that $\overline{X}_{i}\neq\overline{X}_{i+1}.$ This implies that there
exists a maximal subgroup $M$ of $G$ containing $X_{i+1}$ but not $X_{i}$. We
have $X_{i+1}\leq X_{i}\cap M<X_{i}.$ Since $X_{i}\cap M\in\mathcal{F}$ and
$\mathcal{C}$ cannot be refined in $\mathcal{F}$, we deduce $X_{i+1}=X_{i}\cap
M.$ But then $X_{i}/X_{i+1}=X_{i}/(X_{i}\cap M)\cong X_{i}M/M=G/M,$ hence
$X_{i+1}$ is a maximal subgroup of $X_{i}$ (and therefore $\overline{X}_{i+1}$
is a maximal subgroup of $\overline{X}_{i}).$ This implies that the length of
$\overline{\mathcal{C}}$ is precisely $u,$ hence $t\geq u.$ ∎

######  Theorem 17.

Let $G$ be a finite group. If $G/\operatorname{Fit}(G)$ is nilpotent, then

$\operatorname{MinInt}(G)\geq m(G).$

###### Proof.

We may assume $\operatorname{Frat}(G)=1.$ In this case
$\operatorname{Fit}(G)=\operatorname{soc}(G)$ has a complement, say $X$, in
$G$ and $Z(G)$ has a complement, say $W$, in $\operatorname{Fit}G$, which is
normal in $G$. Let $H=Z(G)X.$ We have

$G=W\rtimes H=\left(V_{1}^{\delta_{1}}\times\dots\times
V_{t}^{\delta_{t}}\right)\rtimes H,$

where $V_{1},\dots,V_{t}$ are faithful irreducible $H$-modules, pairwise not
$H$-isomorphic. By [10, Theorem 1], $m(G)$ is the number of non-Frattini
factors in a chief series of $G.$ Hence $m(G)=\delta_{1}+\dots+\delta_{t}+u,$
where $u$ be the composition length of $H/\operatorname{Frat}(H).$ Let
$\mathcal{M}$ be the family of the maximal subgroups of $H$ and let

$\mathcal{D}=\\{C_{H}(v)\mid v\in V_{i},1\leq i\leq t\\}.$

Consider the family $\mathcal{F}$ consisting of $H$ and all the possible
intersections of elements of $\mathcal{D}$ and $\mathcal{M}.$

Now let $1=Y_{\rho}<\dots<Y_{0}=G$ be a chain of maximal intersections in $G$
that cannot be refined. By an iterated application of [7, Theorem 15], there
exists $w\in W$ such that for $1\leq i\leq\rho$, we have
$Y_{i}=U_{i}Z_{i}^{w},$ with $U_{i}\leq_{H}W$ and $Z_{i}\in\mathcal{F}.$
Moreover either $Z_{i}=Z_{i+1}$ and $U_{i}/U_{i+1}\cong_{H}V_{j}$ for some
$1\leq j\leq t$ or $U_{i}=U_{i+1}$ and $Z_{i+1}=Z_{i}\cap X$ with
$X\in\mathcal{D}\cup\mathcal{M}.$ In the second case, the fact that there is
no maximal intersection in $G$ strictly between $Y_{i+1}$ and $Y_{i}$ implies
that $Z_{i+1}$ is $\mathcal{F}$-maximal in $Z_{i}.$ Let $J:=\\{j\mid
U_{j+1}<U_{j}\\}$. Assume $|J|=a$ and order the elements of $J$ so that
$j_{1}<j_{2}<\dots<j_{a}$: we have that
$0<U_{j_{a}}<U_{j_{a-1}}\dots<U_{j_{2}}<U_{j_{1}}=W$ is an $H$-composition
series of $W$ and this implies $a=\delta_{1}+\dots+\delta_{t}.$ Now let
$J^{*}:=\\{j\mid U_{j+1}=U_{j}\\}$. Assume $|J^{*}|=b$ and order the elements
of $J$ so that $j_{1}<j_{2}<\dots<j_{b}$: we have that
$1<U_{j_{b}}<U_{Z_{b-1}}\dots<Z_{j_{2}}<Z_{j_{1}}=H$ is an $\mathcal{F}$-chain
that cannot be refined, so by Lemma 16, $b\geq u$. We conclude
$\rho=a+b\geq\delta_{1}+\dots+\delta_{t}+u=m(G).$ ∎

Remark. One could hope to generalize Lemma 16 as follows: let $G$ be a finite
soluble group and let $\mathcal{F}$ be a family of subgroups of $G$ which
contains $G$ and all the maximal subgroups of $G$ and is closed under taking
intersections. Let $\mathcal{C}=X_{t}<X_{t-1}<\dots X_{1}<X_{0}$ be a chain in
$\mathcal{F}$. If $\mathcal{C}$ cannot be refined in $\mathcal{F}$, then
$t\geq m(G).$ This would allows to prove Theorem 17 for arbitrary finite
soluble groups. However this more general statement is false. Let
$G=H_{1}\times H_{2}\times H_{3},$ with $H_{i}\cong\operatorname{Sym}(3)$ and
let $K_{i}$ be the Sylow 3-subgroup of $H_{i}.$ Let $1\neq k_{i}\in K_{i}$ and
let $X=\langle k_{1},k_{2},k_{3}\rangle.$ Let $\mathcal{F}$ be the family of
subgroups of $G$ consisting of $G,$ the maximal intersections in $G$ and $X.$
A maximal subgroup of $G$ containing $X$, contains also $K:=K_{1}\times
K_{2}\times K_{3},$ so $1<X<K<K_{1}\times K_{2}\times H_{3}<K_{1}\times
H_{2}\times H_{3}=G$ in a non-refinable $\mathcal{F}$-chain in $G$. However
$m(G)=6.$

##  4\. Strongly and weakly minmax finite groups

######  Lemma 18.

Let $N$ be a normal subgroup of a strongly minmax finite group. If $N$ in an
intersection of maximal subgroups of $G$ and $G$ is weakly minmax, then $G/N$
is weakly minmax.

###### Proof.

Let $\mathcal{M}^{+}(G,N)$ (resp. $\mathcal{M}^{-}(G,N)$ be the sublattice of
$\mathcal{M}(G)$ consisting of the subgroups in $\mathcal{M}(G)$ containing
$N$ (resp. contained in $N$). Let $N=H_{0}\leq\dots\leq H_{t}=G$ and
$N=K_{0}\leq\dots\leq K_{u}=G$ be two unrefinable chains in
$\mathcal{M}^{+}(G,N)$. If $T_{0}\leq\dots\leq T_{v}=N$ is an unrefinable
chain in $\mathcal{M}^{-}(G,N),$ then

$1=T_{0}\leq\dots\leq T_{v}=H_{0}\leq\dots\leq H_{t}=G$

and

$1=T_{0}\leq\dots\leq T_{v}=K_{0}\leq\dots\leq K_{u}=G$

are two unrefinable chains in $\mathcal{M}(G)$, so, since $G$ is weakly
minmax, they have the same length, but then $t=u$ and therefore
$\operatorname{MinInt}(G/N)=\operatorname{MaxInt}(G/N).$

Now assume that $X_{1},\dots,X_{a}$ is a family of maximal subgroups of $G$
with minimal size with respect to the property $X_{1}\cap\dots\cap X_{a}=N$
and that $Y_{1},\dots,Y_{b}$ is a family of maximal subgroups of $G$
containing $N$ and with the property that the chain

$\mathcal{C}:Y_{1}\cap\dots\cap Y_{b}<Y_{1}\cap\dots\cap
Y_{b-1}<\dots<Y_{1}\cap Y_{2}<Y_{1}<G$

cannot be refined in $\mathcal{M}^{+}(G,N).$ We want to prove that $a=b.$
First, notice that, since $N$ is a maximal intersection and $\mathcal{C}$ is
not refinable, $Y_{1}\cap\dots\cap Y_{b}=N.$ There exists $Z_{1},\dots,Z_{c}$
such that $X_{1},\dots,X_{a},Z_{1},\dots,Z_{c}$ is a maximal irredundant
family of maximal subgroups of $G$. Consider the chain of maximal
intersections:

$\displaystyle X_{1}\cap\dots\cap X_{a}\cap Z_{1}\cap\dots\cap
Z_{c}<X_{1}\cap\dots\cap X_{a}\cap Z_{1}\cap\dots\cap Z_{c-1}<$
$\displaystyle<\dots<X_{1}\cap\dots\cap X_{a}\cap Z_{1}<X_{1}\cap\dots\cap
X_{a}<\dots<X_{1}<G.$

Since $G$ is weakly minmax, this chain cannot be refined (and in particular
$a+c=\operatorname{MinInt}(G)=\operatorname{MaxInt}(G)$). On the other hand

$\displaystyle N\cap Z_{1}\cap\dots\cap Z_{c}<N\cap Z_{1}\cap\dots\cap
Z_{c-1}<$ $\displaystyle<\dots<N\cap Z_{1}<N=Y_{1}\cap\dots\cap
Y_{b}<\dots<Y_{1}<G$

is a chain of maximal intersections and we must have
$c+b\leq\operatorname{MaxInt}(G)=a+c,$ hence $b\leq a$ (and consequently
$b=a$). ∎

######  Theorem 19.

A finite weakly minmax group is soluble.

Before to prove this theorem, we need to introduce a couple of definitions and
related lemmas.

######  Definition 20.

Let $X$ be an almost simple group and $S=\operatorname{soc}X.$

  1. (1)

We define $\sigma(X)$ as the largest positive integer $\sigma$ for which there
exists a core-free maximal subgroup $Y$ of $X$ and
$s_{1},s_{2},\dots,s_{\sigma}$ in $S$ such that

$Y^{s_{1}}\cap S>Y^{s_{1}}\cap Y^{s_{2}}\cap S>\dots>Y^{s_{1}}\cap
Y^{s_{2}}\cap\dots\cap Y^{s_{\sigma}}\cap S.$

  2. (2)

We define $\tau(X)$ as the minimal size of a family of core-free maximal
subgroups of $X$ with trivial intersection.

######  Lemma 21.

$\sigma(X)\geq 3.$

###### Proof.

Let $Y$ be a core-free maximal subgroup of $X$ and let $T=S\cap Y$. We have
$T\neq 1$ (see for example the last paragraph of the proof of the main theorem
in [13]). If suffices to prove that there exists $s\in S$ such that $1<T\cap
T^{s}<T.$ We have $Y=N_{G}(T)$, so $T=N_{S}(T).$ Assume by contradiction
$1=T\cap T^{s}$ for every $s\in S\setminus T$: this means that $S$ is a
Frobenius group and $T$ is a Frobenius complement, but this is in
contradiction with the fact that $S$ is a non-abelian simple group. ∎

######  Lemma 22.

$\tau(X)\leq 4,$ with equality if and only if $X=U_{4}(2).2$

###### Proof.

See [4, Theorem 1]. ∎

###### Proof of Theorem 19.

Let $G$ a finite weakly minmax group. If $G$ is not soluble, then it admits a
non-abelian chief factor $H/K.$ Let $C=C_{G}(H/K).$ Then $G/C$ is a monolithic
group (with socle isomorphic to $H/K$) and is weakly minmax by Lemma 18. So in
order to complete the proof, it would suffice to prove that a finite
monolithic group with non-abelian socle cannot be weakly minmax.

Let $G$ be a monolithic primitive group, and assume
$N=\operatorname{soc}(G)\cong S^{n},$ with $S$ a non-abelian simple group. Let
$\psi$ be the map from $N_{G}(S_{1})$ to $\operatorname{Aut}(S)$ induced by
the conjugacy action on $S_{1}$. Set $X=\psi(N_{G}(S_{1}))$ and note that $X$
is an almost simple group with socle $S=\operatorname{Inn}(S)=\psi(S_{1})$.
Let $T:=\\{t_{1},\ldots,t_{n}\\}$ be a right transversal of $N_{G}(S_{1})$ in
$G;$ the map

$\phi_{T}:G\to X\wr\operatorname{Sym}(n)$

given by

$g\mapsto(\psi(t_{1}gt_{1\pi_{g}}^{-1}),\dots,\psi(t_{n}gt_{n\pi_{g}}^{-1}))\pi_{g}$

where $\pi_{g}\in\operatorname{Sym}(n)$ satisfies $t_{i}gt_{i\pi_{g}}^{-1}\in
N_{G}(S_{1})$ for all $1\leq i\leq n$, is an injective homomorphism. So we may
identify $G$ with its image in $X\wr T$, where $T=\\{\pi_{g}\mid g\in G\\}$ is
a transitive subgroup of $\operatorname{Sym}(n).$ In this identification, $N$
is contained in the base subgroup $X^{n}$ and $S_{i}$ is a subgroup of the
$i$-th component of $X^{n}$.

Let $F/N=\operatorname{Frat}(G/N)$ and assume that $Y_{1},\dots,Y_{t}$ is a
family of maximal subgroups of $G$ of minimal size with respect to the
property $F=Y_{1}\cap\dots\cap Y_{u}.$ Now choose a core-free maximal subgroup
$Y$ of $X$ and $s_{1},\dots,s_{\sigma}$ as in the definition of
$\sigma=\sigma(X)$ and let $M=G\cap(Y\wr T)$. By [1] Proposition 1.1.44, $M$
is a maximal subgroup of $G$. For $1\leq i\leq n$ and $1\leq j\leq\sigma,$ let
$\tau_{i,j}=(1,\dots,1,s_{j},1\dots,1)\in S^{n},$ where $s_{j}$ is in the
$i$-th position of $\tau_{i,j}$ and let $M_{i,j}=M^{\tau_{i,j}}.$ We order
lexicologically the pairs $(i,j)$. Let $\Sigma_{k,l}=\cap_{1\leq i\leq k,1\leq
j\leq l}M_{i,j}\cap F.$ We have $\Sigma_{k,l}\cap M_{k,l}<\Sigma_{k,l}$ and
this implies

(4.1) $\operatorname{MaxInt}(G)\geq t+n\cdot\sigma(X).$

Now let $\tau=\tau(X)$ and suppose that $R_{1},\dots,R_{\tau}$ are core-free
maximal subgroups of $X$ with trivial intersection. Again by [1] Proposition
1.1.44, $Z_{i}:=G\cap(R_{i}\wr T)$ is a maximal subgroup of $G$ for $1\leq
i\leq\tau.$ Let $W=Y_{1}\cap\dots\cap Y_{t}\cap Z_{1}\cap\dots\cap Z_{\tau}$
and let $\pi:G\to T$ the epimorphism sending $g$ to $\pi_{g}.$ Since $W\cap
X^{n}=1,$ we have $W\cap\ker\pi=1$, so $W$ is isomorphic to a (proper)
subgroup of $T.$ Moreover since $W\cap N=1,$ we have $W\cong WN/N\leq F/N,$
hence $W$ is a nilpotent subgroup of $\operatorname{Sym}(n).$ With the same
arguments used by Cameron, Solomon and Turull in [5], it can be easily proved
that the maximal length $l(K)$ of a chain of subgroups in a nilpotent
permutation group of degree $n$ is at most $n-1.$ It follows $l(W)\leq n-1.$
Since $G$ has trivial Frattini subgroup, there exist at most $n-1$ maximal
subgroups of $G$ whose total intersection with $W$ is trivial, hence

(4.2) $\alpha(G)\leq t+\tau(G)+n-1.$

Since $G$ is weakly minmax, combining (4.1) and (4.2), we get

(4.3) $n\cdot(\sigma(X)-1)<\tau(X)$

By Lemmas 21 and 22, $2n<4,$ hence $n=1.$

We have so proved that $n=1,$ i.e. $G$ is an almost simple group. First assume
that $G=S$ is a simple group. We have
$\alpha(G)=\operatorname{MaxInt}(G)\geq\operatorname{MaxDim}(G)\geq m(G)\geq
3$. Since, by [3, Theorem 1], $\alpha(G)\leq 3,$ it follows
$\operatorname{MaxDim}(G)=m(G)=\alpha(G)=3.$ Since $m(A_{n})\geq n-2$ and
$\alpha(A_{5})=2$, it follows that $G$ is not an alternating group. If $G$ is
sporadic, then by [3, Theorem 3.1] the condition ${\alpha}(G)=3$ implies that
$G={\rm M}_{22}$. However ${\rm M}_{22}$ has a maximal subgroup $H=\rm
L_{3}(4)$ with $b({\rm M}_{22},H)=5$ (see [2, Table 1]) and we deduce that
${\operatorname{MaxInt}}({\rm M}_{22})\geq 5.$ If $G$ is an exceptional group
of Lie type, then [3, Theorem 1] implies that $G=G_{2}(2)^{\prime}\cong{\rm
U}_{3}(3),$ however, by a theorem of Wagner [16], $G$ can be generated by $4$
involutions and no fewer, so $m(G)\geq 4$.

So we may assume $S<G.$ Let $H$ be a core-free maximal subgroup of $G$ and let
$b=b(G,H)$ be the minimal size of a set of conjugates of $H$ with trivial
intersection (i.e. the base size of the primitive action of $G$ on the set of
the right cosets of $H$). This set of conjugates of $H$ is an irredundant
family of maximal subgroups with maximal size, so $\alpha(G)\leq
b(G,H)\leq\operatorname{MaxDim}(G)\leq\operatorname{MaxInt}(G).$ In
particular, if $G$ is weakly minmax, then $b(G,H)$ is the same for any choice
of a core-free maximal subgroup $H$ of $G$. It follows from [2], that if
$\operatorname{soc}G$ is a sporadic simple group, then $G$ has faithful
primitive actions with different base sizes, hence $G$ is not weakly minmax.
In any case, by Lemma 22, if $G$ is weakly minmax then
$\alpha(G)=\operatorname{MaxDim}(G)=\operatorname{MinInt}(G)=\operatorname{MaxInt}(G)\leq\tau(G)=3$
if $G\neq U_{4}(2).2,$
$\alpha(G)=\operatorname{MaxDim}(G)=\operatorname{MinInt}(G)=\operatorname{MaxInt}(G)\leq\tau(G)=4$
if $G=U_{4}(2).2.$

If $n\geq 5,$ then $b(\operatorname{Sym}(n),\operatorname{Sym}(n-1))=n-1\geq
4>3$, and therefore $\operatorname{Sym}(n)$ is not weakly minmax. Finally
assume that $G$ is an almost simple with a socle $S$ of Lie type and $S<G.$
Let $B$ be Borel subgroup of $S$ and let $u$ be the number of the nodes of the
associated Dynkin diagram, or the number of the orbits for a suitable groups
of symmetries of this diagram if $S$ is of twisted type or $G$ involves a
graph automorphism of $G).$ There exists a family $Y_{1},\dots,Y_{u}$ of
maximal parabolic subgroups of $G$ such that $S>S\cap Y_{1}>S\cap Y_{1}\cap
Y_{2}>\cdots>S\cap Y_{1}\cap Y_{2}\cap\dots\cap Y_{u}=B$. Moreover, as in the
proof of Lemma 21, since $N_{S}(B)=B$ and $S$ is not a Frobenius group, there
exists $x\in S$ with $1<B\cap B^{x}<B.$ Let $m=m(G/S)$. There exists an
irredundant family $X_{1},\dots,X_{m}$ of maximal subgroup of $G$ containing
$S$. Let $X=X_{1}\cap\dots\cap X_{m}.$ Then $1<(X\cap Y_{1}\cap\dots\cap
Y_{t})\cap(X\cap Y_{1}\cap\dots\cap Y_{t})^{x}<X\cap Y_{1}\cap\dots\cap
Y_{t}<\dots<X\cap Y_{2}\cap Y_{1}<Y_{1}\cap X<X<\cdots<X_{2}\cap
X_{1}<X_{1}<G$ is a chain in $\mathcal{M}(G)$, so
$\tau(G)\geq\operatorname{MinInt}(G)\geq m+u+2\geq 3+u,$ a contradiction. ∎

######  Proposition 23.

Let $G$ be a primitive monolithic soluble group. If $G$ is weakly minmax, then
the derived length of $G$ is at most 3.

###### Proof.

Assume $G=V\rtimes H,$ where $V$ is an irreducible $H$-module and $H$ is a
finite soluble group. By [15, Theorem 2.1], the base size $b(H)$ of $H$ on $V$
is at most 3, i.e. there exist $v_{1},v_{2},v_{3}\in V$ such that
$C_{H}(v_{1})\cap C_{H}(v_{2})\cap C_{H}(v_{3})=1$. This implies $1=H\cap
H^{v_{1}}\cap H^{v_{2}}\cap H^{v_{3}},$ hence $\alpha(G)\leq 4.$

Let $1=X_{0}<X_{1}<\dots<X_{t}=G$ be a chain of normal subgroups in
$\mathcal{M}(G)$ with the property that, for every $0\leq i\leq t-1,$ there is
no normal subgroup $Y\in\mathcal{M}(G)$ with $X_{i}<Y<X_{i+1}.$ For every
$0\leq i\leq t-1$, let $Y_{i}/X_{i}$ be a minimal normal subgroup of $G/X_{i}$
contained in $X_{i+1}/X_{i}.$ If $M$ is a maximal subgroup of $G$ containing
$Y_{i},$ then $\operatorname{Core}_{G}(M)\cap X_{i+1}\in\mathcal{M}(G)$ is a
normal subgroup containing $Y_{i},$ hence $\operatorname{Core}_{G}(M)\cap
X_{i+1}=X_{i+1}$ and consequently $X_{i+1}\leq M.$ This implies
$X_{i+1}/Y_{i}=\operatorname{Frat}(G/Y_{i}).$ In particular $Y_{0}=V$ and
$X_{1}=V\operatorname{Frat}(H).$ If we refine the normal series
$X_{0}<Y_{0}<X_{1}<Y_{1}<\dots<X_{t}=G$ to a chief series of $G,$ the non-
Frattini factors are precisely the $t$ factors $Y_{i}/X_{i}$ for $0\leq i\leq
t-1.$ In particular, by [10, Theorem 2], $t=m(G).$ Since $G$ is weakly minmax,
$t=m(G)\leq\operatorname{MaxDim}(G)\leq\operatorname{MaxInt}(G)=\alpha(G)\leq
4.$

Assume $t=4.$ In this case $\alpha(G)=m(G)=4$ and the chain
$1<X_{1}<X_{2}<X_{3}<X_{4}=G$ cannot be refined inserting other maximal
intersections. In particular $X_{1}\cap H=1$ and therefore $Y_{1}=X_{1}=V$ and
$\operatorname{Frat}(H)=1.$ If $i\geq 1,$ then $X_{i}=VZ_{i}$, with
$Z_{i}\in\mathcal{M}(H)$ and $1=Z_{1}=H\cap X_{1}<Z_{2}=H\cap
X_{2}<Z_{3}=H\cap X_{3}<Z_{4}=H<G$ is a non refinable chain in
$\mathcal{M}(G).$ Since $Z_{2}$ is normal in $H$ and $V$ is a faithful
irreducible $H$-module, we must have $C_{V}(Z_{2})=1.$ Let $0\neq v\in V.$
Then $C_{Z_{2}}(v)=H^{v}\cap Z_{2}<Z_{2}$ and therefore, since $H^{v}$ is a
maximal subgroup of $G$, we must have $C_{Z_{2}}(v)=1$. In particular if $N$
is a minimal normal subgroup of $H$ contained in $Z_{2},$ then $N$ is an
elementary abelian group acting fixed-point-freely on $V$, so it is cyclic of
prime order (see [14, 10.5.5]). Moreover $H$ is not a supersoluble, otherwise
there would $x$ and $y$ in $V$ such that $C_{H}(x)\cap C_{H}(y)=1$ (see [17,
Theorem A]) and consequently $\alpha(G)\leq 3.$ In particular $G$ contains a
minimal normal subgroup $M$ which is not $H$-isomorphic to $N.$ Since
$m(H)=m(G)-1=3,$ we must have $H\cong(N\times M)\rtimes K$ with $K$ a cyclic
group of order $p^{t}$ for a suitable prime $p.$ In particular
$V<VMK^{p}<VNMK^{p}<VH=G$ is a chain of normal subgroups in $\mathcal{M}(G)$.
Arguing as before, we deduce that also $M$ acts fixed-point-freely of $V$, but
this would imply that $M$ is cyclic of prime order and $H$ is supersoluble.
This excludes $t=4.$

Assume $t=3.$ Then $m(H)=3$, so in particular there exist two primes $p$ and
$q$ and a normal subgroup $N$ of $H$ such that $F=\operatorname{Frat}(H)\leq
N,$ $N/F$ is a non-Frattini chief factor of $H$ of $q$-power order and $H/N$
is cyclic of $p$-power order. Let $K$ be the unique maximal subgroup of $H$
containing $N.$ Then $1\leq F<VF<VK<G$ is a chain in $\mathcal{M}(H),$ so
$\operatorname{MaxInt}(G)\geq 3,$ with equality only if $F=1$. In particular
if $\alpha(G)=3$, then $F=1$ and $G$ has derived length at most 3. We remain
with the case $F\neq 1$ and $\alpha(G)=4.$ This implies in particular that the
minimal size $b(H)$ of $H$ on $V$ is equal to 3. By [9, Theorem 1.1],
$(|H|,|V|)\neq 1.$ If follows that $V$ is a $q$-group and $F$ a $p$-group. Let
$P$ be a Sylow $q$-subgroup of $H$. Since $P$ is contained in $N$, by the
Frattini Argument, $G=NN_{G}(P)=FPN_{G}(P)=FN_{G}(P)=N_{G}(P)$ so $P$ is
normal in $G$ and $\operatorname{Frat}(G/P)\geq PF/P=N/P$ ([14, 5.2.13
(iii)]). But then $G/P$ is a cyclic $p$-group. This would imply that $H$ is
supersoluble, and consequently $b(H)=2$ by [17, Theorem A], a contradiction.

Finally, if $t\leq 2,$ then $m(H)\leq 1$ and consequently $H$ is cyclic and
$G$ is metabelian. ∎

######  Corollary 24.

If $G$ is weakly minmax, then the derived length of $G/\operatorname{Frat}(G)$
is at most 3.

###### Proof.

It follows immediately from the fact that $G$ can be embedded in
$\prod_{N\in\Omega}G/N$ being $\Omega$ be the family of the normal cores of
the maximal subgroups of $G$ and that $G/N$ has derived length at most 3 for
any $N\in\Omega$ by the previous proposition. ∎

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