# Finite solvable tidy Groups whose orders are divisible by two primes

Nicolas F. Beike  Department of Mathematical Sciences, Kent State University,
Kent, OH 44242<EMAIL_ADDRESS>,  Rachel Carleton  Department of Mathematical
Sciences, Kent State University, Kent, OH 44242<EMAIL_ADDRESS>,  David G.
Costanzo  School of Mathematical and Statistical Sciences, O-110 Martin Hall,
Box 340975, Clemson University, Clemson, SC 29634<EMAIL_ADDRESS>,
Colin Heath  New York University School of Law, 40 Washington Square South,
New York, NY 10012<EMAIL_ADDRESS>,  Mark L. Lewis  Department of
Mathematical Sciences, Kent State University, Kent, OH 44242
<EMAIL_ADDRESS>,  Kaiwen Lu  Department of Mathematics, Brown University,
Providence, RI 02912<EMAIL_ADDRESS>and  Jamie D. Pearce  Department
of Mathematics, University of Texas at Austin, 2515 Speedway, PMA 8.100,
Austin, TX 78712<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate finite solvable tidy groups. We classify the
tidy $\\{p,q\\}$-groups. Combining this with a previous result, we are able to
characterize the finite tidy solvable groups. Using this characterization, we
bound the Fitting height of finite tidy solvable groups and we prove that the
quotients of finite tidy solvable groups are tidy.

###### Key words and phrases:

tidy groups, $\\{p,q\\}$-groups, solvable groups

###### 2010 Mathematics Subject Classification:

Primary: 20D10 Secondary: 20D20

##  1\. Introduction

Throughout this paper, all groups are finite except where stated. We use [11]
for standard group theory results; but any standard group theory text will
contain nearly all of the results that we need. For a group $G$ and an element
$x\in G$, we condier the set ${\rm Cyc}_{G}(x)=\\{g\in G\mid\langle
x,g\rangle{\rm~{}is~{}cyclic}\\}$. For most groups $G$ and elements $x$, this
set ${\rm Cyc}_{G}(x)$ is not a subgroup. In this paper, we focus on the
following special situation. As in [12], a group $G$ is said to be tidy if
${\rm Cyc}_{G}(x)$ is a subgroup of $G$ for every element $x\in G$.

Using the definition of tidy, it is an easy exercise to show that subgroups of
tidy groups are tidy. On the other hand, in [12], it was shown that if $G$ is
nilpotent and the Sylow subgroups of $G$ are tidy, then $G$ is tidy. In [5],
we determine that a $p$-group is tidy if and only if it is cyclic, has
exponent $p$, is dihedral, or is generalized quaternion. Therefore, we know
all of the nilpotent tidy groups.

It is natural next to look at solvable groups. First, one would ask if there
are non-tidy solvable groups where all of the Sylow subgroups are tidy. In
[6], we show that $S_{3}\times Z_{3}$ is an example of a solvable group where
all of the Sylow subgroups are tidy but the group itself is not tidy. However,
when we consider Hall subgroups with two prime divisors, we obtain a different
answer. The main result in [6] shows that if $G$ is a solvable group and every
Hall subgroup whose order is divisible by two prime divisors is tidy, then $G$
is tidy.

Thus, if we can classify the $\\{p,q\\}$-groups that are tidy for primes $p$
and $q$, then we can use the main result from [6] to characterize the solvable
tidy groups. In fact, we are able to classify the tidy $\\{p,q\\}$-groups in
Theorem 4.6. Due to the complexity of the list of groups, we will leave the
presentation of this classification for Section 4. We now present two
consequences for solvable tidy groups that we can obtain from our
characterization.

First, we are able to bound the Fitting height of the group and the derived
length of the quotient modulo the Fitting subgroup.

######  Theorem 1.

Let $G$ be a solvable, tidy group. Then $G$ has Fitting height at most $4$ and
$G/F(G)$ has derived length at most $4$. If $|G|$ is odd, then $G$ has Fitting
height at most $3$ and $G/F(G)$ is abelian or metabelian.

We note that the definition of tidy groups is not particularly compatible with
quotients. It is noted by Erfanian and Farrokhi in the Introduction of [8]
that the quotients of infinite tidy groups are not necessarily tidy. They give
an example of an infinite tidy group with a quotient that is not tidy.

We now give some evidence that for finite tidy groups, quotients of tidy
groups are tidy. In particular, we prove that the quotients of finite solvable
tidy groups are tidy.

######  Theorem 2.

If $G$ is a (finite) solvable tidy group and $N$ is a normal subgroup of $G$,
then $G/N$ is tidy.

As the above shows, one of our methods of analyzing tidy groups is in terms of
their Hall subgroups. In order to obtain many of our results, we also need a
second method of analysis. This method involves studying the elements of prime
power order. We will show in Section 2 that it is sufficient to check whether
the elements of prime power order satisfy the condition of the definition to
determine if a group is tidy.

This research was conducted during a summer REU in 2020 at Kent State
University with the funding of NSF Grant DMS-1653002. We thank the NSF and
Professor Soprunova for their support.

##  2\. Elements of Prime Power Order

In [6], we prove the following result which shows that need only check on the
elements of prime power order.

######  Lemma 2.1.

Suppose $G$ is a group. If every element $1\neq x\in G$ having prime power
order satisfies the condition that ${\rm Cyc}_{G}(x)$ is a subgroup of $G$,
then $G$ is a tidy group.

With this in mind, we can focus on the elements of prime power order. Since
subgroups of tidy groups are tidy, we need to focus on tidy Sylow
$p$-subgroups. In Theorem 14 of [12], they obtain a characterization of the
tidy $p$-groups. In [5], we use their characterization to obtain a
classification of the tidy $p$-groups.

######  Theorem 2.2.

Let $G$ be a $p$-group for some prime $p$. Then $G$ is a tidy group if and
only if one of the following occurs:.

  1. (1)

$G$ has exponent $p$.

  2. (2)

$G$ is cyclic.

  3. (3)

$p=2$ and $G$ is dihedral or generalized quaternion.

The following is a consequence of Burnside’s normal $p$-complement theorem
(see Theorem 5.13 of [11]) and Fitting’s theorem (see Theorem 4.34 of [11]).
It is proved as Lemma 2.5 of [6].

######  Lemma 2.3.

Let $G$ be a group and let $p$ be a prime. If $G$ has a cyclic Sylow
$p$-subgroup and $p$ divides $|Z(G)|$, then $G$ has a normal $p$-complement.

When a Sylow $p$-subgroup of $G$ is cyclic or generalized quaternion and $x\in
Z(G)$ has $p$-power order, then ${\rm Cyc}_{G}(x)$ is a subgroup. This was
proved in Lemma 2.6 of [6].

######  Lemma 2.4.

Let $G$ be a group and let $p$ be a prime. Suppose $1\neq x\in Z(G)$ has
$p$-power order. If a Sylow $p$-subgroup of $G$ is either cyclic or
generalized quaternion, then $G={\rm Cyc}_{G}(x)$.

For central elements of order $p$ when the Sylow subgroup has exponent $p$, we
proved the following in Lemma 2.4 of [6].

######  Lemma 2.5.

Let $G$ be a group and let $p$ be a prime. Suppose $x\in Z(G)$ has order $p$
and a Sylow $p$-subgroup of $G$ has exponent $p$. Then ${\rm Cyc}_{G}(x)$ is a
subgroup of $G$ if and only if $G$ has a normal $p$-complement. In this case,
${\rm Cyc}_{G}(x)=\langle x\rangle K$ where $K$ is the normal $p$-complement
of $G$.

For a central element of order $2$ in a group whose Sylow $2$-subgroups are
dihedral and there is a normal $2$-complement, we proved the following as
Lemma 2.7 of [6]. It made use of a result proved by Gorenstein and Walter in
[9] regarding groups that have a dihedral subgroup as a Sylow $2$-subgroup.

######  Lemma 2.6.

Let $G$ be a group and let $x\in Z(G)$ have order $2$. Suppose a Sylow
$2$-subgroup $T$ of G is dihedral and write $D$ for the cyclic subgroup of
index $2$ in $T$. Then $G$ has a normal $2$-complement $K$ and ${\rm
Cyc}_{G}(x)=DK$. In particular, ${\rm Cyc}_{G}(x)$ is a subgroup of G.

The following lemma is key to understanding how coprime subgroups interact in
a tidy group.

######  Lemma 2.7.

Let $G$ be a tidy group, and let $p$ be a prime. If $O_{p}(G)>1$ and $x\in G$
has order not divisible by $p$, then one of the following occurs:

  1. (1)

$x$ centralizes $O_{p}(G)$.

  2. (2)

$\langle x\rangle O_{p}(G)/C_{\langle x\rangle}(O_{p}(G))$ is a Frobenius
group with Frobenius kernel $O_{p}(G)\cong O_{p}(G)C_{\langle
x\rangle}(O_{p}(G))/C_{\langle x\rangle}(O_{p}(G))$.

  3. (3)

$p=2$, the subgroup $O_{2}(G)$ is the quaternion group of order $8$, and
$\langle x\rangle O_{2}(G)/C_{\langle x\rangle}(O_{2}(G))$ is isomorphic to
${\rm SL}_{2}(3)$.

###### Proof.

Without loss of generality, we may assume that $G=\langle x\rangle O_{p}(G)$.
If $x$ centralizes $O_{p}(G)$, then we are done. Thus, we may assume that $x$
does not centralize $O_{p}(G)$. We know via Theorem 2.2 that a Sylow
$p$-subgroup is either cyclic, dihedral, generalized quaternion, or has
exponent $p$. It is not difficult to see that this implies that $O_{p}(G)$
must also be in this same list. If $O_{p}(G)$ is dihedral, generalized
quaternion but not quaternion, or cyclic of $2$-power order, then we know that
${\rm Aut}(O_{p}(G))$ is a $2$-group and we conclude that $x$ must centralize
$O_{p}(G)$. Suppose that $O_{2}(G)$ is the quaternion group of order $8$. In
this case, we know that ${\rm Aut}(O_{2}(G))$ is isomorphic to $S_{4}$. This
implies that $xC_{\langle x\rangle}(O_{2}(G))$ has order $3$ and $\langle
x\rangle O_{2}(G)/C_{\langle x\rangle}(O_{2}(G))$ is isomorphic to ${\rm
SL}_{2}(3)$.

We now may assume that either $p$ is odd or $O_{p}(G)$ is elementary abelian
if $p=2$. Hence, either $O_{p}(G)$ is cyclic or has exponent $p$. Suppose that
there exists $1\neq z\in Z(O_{p}(G))$ with $C_{\langle x\rangle}(z)>C_{\langle
x\rangle}(O_{p}(G))$. By Lemmas 2.5 or 2.3, $C_{G}(z)$ has a normal
$p$-complement. Observe that $O_{p}(G)$ is contained in $C_{G}(z)$. Now,
$C_{\langle x\rangle}(z)$ and $O_{p}(G)$ are contained in normal subgroups of
$C_{G}(z)$ with coprime orders. This implies that $C_{\langle x\rangle}(z)$
centralizes $O_{p}(G)$ which is a contradiction. Thus, we can conclude that
$C_{\langle x\rangle}(z)=C_{\langle x\rangle}(O_{p}(G))$ for all $z\in
Z(O_{p}(G))$. (As $z\in O_{p}(G)$, the inclusion $C_{\langle
x\rangle}(O_{p}(G))\leq C_{\langle x\rangle}(z)$ holds, and we have just shown
that the containment cannot be proper.)

Now, suppose $1\neq y\in O_{p}(G)$. We see that $Z(O_{p}(G))\leq C_{G}(y)$. By
Lemmas 2.5 or 2.3 again, we see that $C_{G}(y)$ has a normal $p$-complement.
This implies that $C_{\langle x\rangle}(y)$ must centralize $Z(O_{p}(G))$.
Thus, $C_{\langle x\rangle}(y)\leq C_{\langle x\rangle}(z)\leq C_{\langle
x\rangle}(O_{p}(G))$ for all $1\neq y\in O_{p}(G)$. We deduce that the group
$\langle x\rangle O_{p}(G)/C_{\langle x\rangle}(O_{p}(G))$ is a Frobenius
group with Frobenius kernel $O_{p}(G)\cong O_{p}(G)C_{\langle
x\rangle}(O_{p}(G))/C_{\langle x\rangle}(O_{p}(G))$, and the result is proved.
∎

We will see that we need to understand the centralizer of $O_{p}(G)$ in a Hall
$p$-complement. In this next lemma, we consider this situation when a Sylow
$p$-subgroup has exponent $p$.

######  Lemma 2.8.

Let $G$ be a tidy group and let $p$ be a prime. Suppose that $G$ has a Sylow
$p$-subgroup $P$ of exponent $p$ and let $H$ be a Hall $p$-complement of $G$.
If $O_{p}(G)>1$ and $C=C_{H}(O_{p}(G))$, then $C$ is normal in $G$. In
addition, if $C<H$, then $O_{p}(G)H/C$ is a Frobenius group with Frobenius
kernel $O_{p}(G)\cong O_{p}(G)C/C$ and Frobenius complement $H/C$ and $P\cap
Z(G)=1$.

###### Proof.

By order considerations, $G=PH$. Since $1<O_{p}(G)$ is a normal subgroup of
$P$, the intersection $O_{p}(G)\cap Z(P)>1$. Thus, we can fix an element $z\in
Z(P)\cap O_{p}(G)$. Observe that $C$ and $P$ are both contained in $C_{G}(z)$.
By Lemma 2.5, we see that $C_{G}(z)$ has a normal $p$-complement. Observe that
$C_{G}(O_{P}(G))\leq C_{G}(z)$, and so, $C_{G}(O_{p}(G))$ has a normal
$p$-complement. Since $O_{p}(G)$ is normal in $G$, we see that
$C_{G}(O_{p}(G))$ is normal. Thus, $C=H\cap C_{G}(O_{p}(G))$ is a Hall
$p$-complement in $C_{G}(O_{p}(G))$. It follows that $C$ is normal in
$C_{G}(O_{p}(G))$, and thus, $C$ is characteristic in $C_{G}(O_{p}(G))$. Using
the fact that $C_{G}(O_{p}(G))$ is normal in $G$, we conclude that $C$ is
normal in $G$.

We now assume that $C<H$. By Lemma 2.7, we see that every element in
$H\setminus C$ acts Frobeniusly on $O_{p}(G)$. It follows that $HO_{p}(G)/C$
is a Frobenius group with Frobenius kernel $O_{p}(G)\cong O_{p}(G)C/C$ and
Frobenius complement $H/C$. Notice that $Z(G)\cap P$ is a normal $p$-subgroup
of $G$, so it is contained in $O_{p}(G)$. Since $O_{p}(G)\cong O_{p}(G)C/C$
and $O_{p}(G)C/C$ is a Frobenius kernel, we conclude that $Z(G)\cap P=1$. ∎

In this next lemma, we make use of a result proved by Gorenstein and Walter in
[9] regarding groups that have a dihedral subgroup as a Sylow $2$-subgroup.

######  Lemma 2.9.

Let $G$ be a group. Suppose a Sylow $2$-subgroup of $G$ is a dihedral group.
If $O_{2}(G)>1$ and is not elementary abelian of order $4$, then $G$ has a
normal $2$-complement.

###### Proof.

Let $P$ be a Sylow $2$-subgroup of $G$. We know that $P$ is a dihedral group
and $O_{2}(G)$ is a normal subgroup of $P$. Since $O_{2}(G)$ is not elementary
abelian of order $4$, we see that $O_{2}(G)$ is either cyclic or dihedral. In
either case, $O_{2}(G)$ has a subgroup $Z$ of order $2$ that is characteristic
in $O_{2}(G)$. It follows that $Z$ is normal in $G$. Hence, $G$ is the
centralizer of an involution. Since a Sylow $2$-subgroup of $G$ is dihedral,
we can apply Lemma 8 of [9] to see that $G$ has a normal $2$-complement. ∎

We continue to investigate tidy groups that have a Sylow $2$-subgroup that is
dihedral. While we are not assuming that $G$ is solvable, we do assume that
$G$ has a Hall $2$-complement.

######  Lemma 2.10.

Let $G$ be a tidy group. Suppose a Sylow $2$-subgroup of $G$ is a dihedral
group. Let $H$ be a Hall $2$-complement of $G$ and let $C=C_{H}(O_{2}(G))$. If
$O_{2}(G)>1$, then $C$ is normal in $G$ and either $G$ has a normal
$2$-complement or $G/C\cong S_{4}$.

###### Proof.

When $O_{2}(G)$ is not elementary abelian of order $4$, we may use Lemma 2.9
to see that $G$ has a normal $2$-complement. Suppose $O_{2}(G)$ is elementary
abelian of order $4$. Write $P$ for a Sylow $2$-subgroup of $G$. Since
$O_{2}(G)$ is a normal subgroup of $P$, we deduce that $P$ is dihedral of
order $8$ and $C_{P}(O_{2}(G))=O_{2}(G)$. In particular, $C_{G}(O_{2}(G))$ has
an elementary abelian Sylow $2$-subgroup. By Lemma 2.8, we see that $C$ is a
normal $2$-complement of $C_{G}(O_{2}(G))$. We see that $G/C$ is isomorphic to
a subgroup of ${\rm Aut}(O_{2}(G))\cong S_{4}$. Since a Sylow $2$-subgroup of
$G$ is dihedral, the result now follows. ∎

##  3\. Some tidy groups

In this section, we prove that several groups are tidy groups. The following
generalizes Proposition 2.5 of [1]. That result proves that a Frobenius group
is tidy if and only if the Frobenius kernel is tidy. We now consider certain
extensions of Frobenius groups to determine when they are tidy.

######  Theorem 3.1.

Let $G$ be a group and let $N<F(G)$ be a normal subgroup of $G$ so that $G/N$
is a Frobenius group with Frobenius kernel $F(G)/N$ and Frobenius complement
$H/N$. Assume that $N$ is a Hall subgroup of $F(G)$ and either (1) $H$ is
nilpotent or (2) $N$ is a Hall subgroup of $H^{\prime}N$, $H^{\prime}N/N$ has
a complement $U/N$ in $H/N$, and $H^{\prime}$ and $U$ are nilpotent groups. If
the Sylow $p$-subgroups of $G$ are tidy for every prime $p$ that divides
$|F(G)|$, then $G$ is tidy.

###### Proof.

By Lemma 2.1, to prove that $G$ is tidy, it suffices to prove that ${\rm
Cyc}_{G}(x)$ is a subgroup for every element $1\neq x\in G$ that has prime
power order. Assume $x\in G$ has order that is a power of the prime $p$.

Suppose first that $p$ does not divide $|F(G)|$. Let $P$ be a Sylow
$p$-subgroup of $G$. We see that $P\cap N\leq P\cap F(G)=1$, so $P\cong PN/N$.
This implies that $P$ is isomorphic to a Sylow subgroup of a Frobenius
complement. Hence, $P$ is either cyclic or generalized quaternion. By Lemma
2.4, we see that ${\rm Cyc}_{C_{G}(x)}(x)=C_{G}(x)$. Since ${\rm
Cyc}_{G}(x)\subseteq C_{G}(x)$, we deduce that ${\rm Cyc}_{G}(x)=C_{G}(x)$ is
a subgroup of $G$.

We now suppose that $p$ divides $|F(G)|$. We are assuming that $G$ has tidy
Sylow subgroups for every prime dividing $|F(G)|$. This implies that all of
the Sylow subgroups of $F(G)$ are tidy. Since $F(G)$ is nilpotent, we conclude
that $F(G)$ is tidy. We next suppose that $p$ does not divide $|N|$. This
implies that $xN$ is a nontrivial element of $F(G)/N$. Since $G/N$ is a
Frobenius group, we deduce that $C_{G/N}(xN)\leq F(G)/N$. It is not difficult
to see that $C_{G}(x)N/N\leq C_{G/N}(xN)$. It then follows that $C_{G}(x)\leq
F(G)$. We now have that ${\rm Cyc}_{G}(x)={\rm Cyc}_{F(G)}(x)$. Since $F(G)$
is tidy, we conclude that ${\rm Cyc}_{G}(x)$ is a subgroup of $F(G)$ and
hence, it is a subgroup of $G$.

We next consider that $p$ divides $|N|$. If a Sylow $p$-subgroup of $C_{G}(x)$
is either cyclic or generalized quaternion, then we know that ${\rm
Cyc}_{G}(x)=C_{G}(x)$ is a subgroup by Lemma 2.4. Thus, we may assume that a
Sylow $p$-subgroup of $C_{G}(x)$ either has exponent $p$ or is dihedral. In
view of Lemmas 2.5 and 2.6, to show that ${\rm Cyc}_{G}(x)$ is a subgroup, it
suffices to show that $C_{G}(x)$ has a normal $p$-complement.

Let $\sigma$ be the set of prime divisors of $|F(G):N|$, and let $L$ be the
Hall $\sigma$-subgroup of $F(G)$. Note that $L=O_{\sigma}(F(G))$, and so $L$
is normal in $G$. Observe that $G=F(G)H=LNH=LH$. Consequently, $G/L\cong
H/(H\cap L)$.The subgroups $H/N$ and $F(G)/N$ intersect trivially, and so
$H\cap F(G)=N$. By Dedekind’s Lemma,

$N=H\cap F(G)=H\cap LN=(H\cap L)N.$

In particular, $(H\cap L)\leq N$. Since $N$ is Hall subgroup of $F(G)$ by our
hypothesis and $|L|$ is a $\sigma$-number, we know that $N\cap L=1$ by order
considerations. Now, $(H\cap L)\leq(N\cap L)=1$, and so $G/L\cong H$.

We proceed with the hypothesis that $H$ is not nilpotent. So, the hypotheses
in (2) are therefore in effect. As $N$ is a normal Hall subgroup of
$H^{\prime}N$, we know that $N\cap H^{\prime}$ is a Hall subgroup of the
nilpotent group $H^{\prime}$. Let $L$ be the complement to $N\cap H^{\prime}$
in $H^{\prime}$, and note that, in fact, $L=O_{\pi}(H^{\prime})$, where $\pi$
is the set of prime divisors of $|H^{\prime}:N\cap
H^{\prime}|=|H^{\prime}N:N|$. So, $L$ is normal in $H$ as $L$ is
characteristic in the subgroup $H^{\prime}$, which in turn is normal in $H$.
(In fact, $L$ is a characteristic subgroup of $H$.) Now,

$H=H^{\prime}NU=H^{\prime}U=L(N\cap H^{\prime})U=LU.$

Since $U/N$ is a complement to $H^{\prime}N/N$, we know that the subgroups
$U/N$ and $H^{\prime}N/N$ intersect trivially. Thus, $U\cap H^{\prime}N=N$.
Note that $H^{\prime}N=L(N\cap H^{\prime})N=LN$, and so by Dedekind’s Lemma

$N=U\cap LN=(U\cap L)N.$

Hence, $(U\cap L)\leq N$, and so $(U\cap L)\leq(N\cap L)=1$, where we used the
fact that $|L|$ and $|N|$ are coprime. Thus, $H/L\cong U$. By (2), $U$ is
nilpotent; hence, $H/L$ is nilpotent. Now, bearing in mind that $L$ is a
$p^{\prime}$-subgroup of $H$, we have that the subgroup $V$ is a normal
$p$-complement of $H$, where $V/L=O_{p^{\prime}}(H/L)$. ∎

In the hypotheses of Theorem 3.1, we have a nilpotent normal subgroup $N$ of a
group $G$ such that $G/N$ is a Frobenius group with Frobenius kernel $F(G)/N$.
Since the Frobenius kernel of a Frobenius group is the Fitting subgroup of the
group, we actually have that

$F(G/N)=F(G)/N.$

Nilpotent normal subgroups $N$ of a group $G$ that satisfy the equation
$F(G/N)=F(G)/N$ were called generalized Frattini subgroups by Beidleman and
Seo in [2] and further studied by Beidleman in [3]. Beidleman and Dykes
considered a further generalization of these subgroups in [4].

It is well known that ${\rm SL}_{2}(3)$ has a unique non split extension by
$Z_{2}$. This group is isoclinic to ${\rm GL}_{2}(3)$ and so, we denote it by
$\widetilde{{\rm GL}_{2}(3)}$. This group occurs as a Frobenius complement and
it has a Sylow $2$-subgroup that is generalized quaternion of order $16$. We
determine the tidiness of certain extensions of $S_{4}$.

######  Theorem 3.2.

Suppose $G$ is a $\\{2,3\\}$-group that satisfies one of the following:

  1. (1)

$O_{2}(G)$ is a Klein $4$-group, $G/O_{3}(G)\cong S_{4}$ and $G/O_{2}(G)$ is a
Frobenius group whose Frobenius kernel is the Sylow $3$-subgroup of
$G/O_{2}(G)$ and whose Frobenius complement has order $2$.

  2. (2)

$O_{2}(G)$ is the Sylow $2$-subgroup of $G$ and is the quaternion group of
order $8$ and $G/O_{3}(G)\cong{\rm SL}_{2}(3)$.

  3. (3)

$O_{2}(G)$ is the quaternion group of order $8$,
$G/O_{3}(G)\cong\widetilde{{\rm GL}_{2}(3)}$ and $G/O_{2}(G)$ is a Frobenius
group whose Frobenius kernel is the Sylow $3$-subgroup of $G/O_{2}(G)$ and
whose Frobenius complement has order $2$.

Then $G$ is a tidy group.

###### Proof.

By Lemma 2.1, it suffices to prove that ${\rm Cyc}_{G}(x)$ is a subgroup for
$1\neq x\in G$ when $x$ has either $2$-power or $3$-power order. Suppose first
that $1\neq x$ has $3$-power order, and let $P$ be a Sylow $3$-subgroup of
$G$. We claim that $C_{G}(x)\leq PO_{2}(G)$. In (2), we have $G=PO_{2}(G)$, so
the claim holds trivially. In (1) and (3), we know $G/O_{2}(G)$ is a Frobenius
group whose Frobenius kernel is $PO_{2}(G)/O_{2}(G)$. This implies that
$C_{G/O_{2}(G)}(xO_{2}(G))\leq PO_{2}(G)/O_{2}(G)$, and so, $C_{G}(x)\leq
PO_{2}(G)$. It follows that the claims holds in these cases also. Thus,
$C_{G}(x)$ has a normal $2$-complement. Since either $P$ is cyclic or has
exponent $3$, this implies ${\rm Cyc}_{G}(x)$ is a subgroup by either Lemma
2.5 or 2.4.

Now, suppose $1\neq x$ has $2$-power order. If $o(x)=4$, then $\langle
x\rangle$ will be a Sylow $2$-subgroup of $C_{G}(x)$, and so, ${\rm
Cyc}_{G}(x)$ is a subgroup by Lemma 2.4. We suppose $o(x)=2$. If a Sylow
$2$-subgroup of G is generalized quaternion, then we have that ${\rm
Cyc}_{G}(x)$ is a subgroup by Lemma 2.4 again. We are left with the case where
a Sylow $2$-subgroup of G is dihedral of order $8$. This implies that we are
in hypothesis (1). It is not difficult to see that $C_{G}(x)$ will contain a
Sylow $2$-subgroup $T$ of $G$. Since $G/O_{3}(G)\cong S_{4}$, we see that
$O_{3}(G)$ is the Sylow $3$-subgroup of $C_{G}(x)$. In particular, $C_{G}(x)$
has a normal $2$-complement. This implies that $C_{G}(x)$ is a subgroup by
Lemma 2.6, and the result is proved. ∎

Recall that a group $G$ is a $2$-Frobenius group if there exists normal
subgroups $1<K<N<G$ so that $N$ is a Frobenius group with Frobenius kernel $K$
and $G/K$ is a Frobenius group with Frobenius kernel $N/K$. The group $S_{4}$
is an example of a $2$-Frobenius group. One consequence of this next lemma is
that $S_{4}$ is the only $2$-Frobenius group that is a tidy group.

######  Lemma 3.3.

Suppose $G$ is a tidy group, and suppose there exists a normal subgroup $L$ so
that $G/L$ is a 2-Frobenius group. If there exists a normal subgroup $M$ of
$G$ so that $L\cap M=1$ and $ML/L$ is the Fitting subgroup of $G/L$, then $M$
is a Klein $4$-subgroup, $G/L$ is isomorphic to $S_{4}$, and $|L|$ is odd.

###### Proof.

We have $1\leq L<K<N<G$ so that $N/L$ is a Frobenius group with Frobenius
kernel $K/L$ and $G/K$ is a Frobenius group with Frobenius kernel $N/K$.
Observe that $K/L$ is the Fitting subgroup of $G/L$, so $K=ML$. Since $M\cap
L=1$, we have $K=M\times L$. Let $H/L$ be a Frobenius complement of $N/L$. By
the Frattini argument, we have that
$G/L=N/LN_{G/L}(H/L)=KH/LN_{G/L}(H/L)=K/LN_{G/L}(H/L)$. Observe that $K/L\cap
N_{G/L}(H/L)=1$, so $N_{G/L}(H/L)\cong G/K$ is a Frobenius group with
Frobenius kernel $H/L$. Let $B/L$ be a Frobenius complement of $N_{G/L}(H/L)$.

Let $p$ be a prime divisor of $|M|$, and let $P$ be a Sylow $p$-subgroup of
$M$. Observe that $PHB/L$ is a 2-Frobenius group. Let $Q/L$ be a subgroup of
$B/L$ of prime order $q$. Observe that $PHQ/L$ is a 2-Frobenius group. Suppose
$p\neq q$. Observe that $HQ/L$ acts nontrivially on $P$ and $Q/L$ cannot be
contained in the kernel of this action. We have that $Q/L$ acts nontrivially
on $P$. On the other hand, $HQ/L$ is not a Frobenius complement, so we have
that $C_{P}(Q/L)>1$. We now see that $C_{Q/L}(P)=1$ and $Q/L$ does not act
Frobeniusly on $P$. This contradicts Lemma 2.7. Thus, we must have $p=q$,
which implies that $M$ and $B/L$ are both $p$-groups. Since $MH/L$ is a
Frobenius group, we see that $M$ is not dihedral or generalized quaternion.
Since $HB/L$ is nonabelian and is isomorphic to a subgroup of
$G/C_{G}(M)\leq{\rm Aut}(M)$, it follows that $M$ must be cyclic.

By Theorem 2.2, $M$ must have exponent $p$. This implies that $Z:=Z(M)$ is
elementary abelian, and we can view $Z$ as a module for $Z_{p}$. Observe also
that $ZHQ/L$ is a $2$-Frobenius group. Now, we now apply Theorem 15.16 of [10]
to see that $Z$ has a basis which is permuted by $Q/L$ in orbits of size $p$.
Let $\\{a_{1},\dots a_{p}\\}$ be one of these orbits. We see that $\langle
a_{1},Q/L\rangle$ is isomorphic to $Z_{p}\wr Z_{p}$. It is well-known that
$Z_{p}\wr Z_{p}$ has exponent $p^{2}$ (see Problem 4A.7 in [11]). Since $M$ is
not cyclic, we can use Theorem 2.2 to see that a Sylow $p$-subgroup of $G$
must have exponent $p$ when $p$ is odd. Hence, we must have that $p=2$. Now,
the Sylow $2$-subgroup of G is not abelian, and has a normal subgroup that is
elementary abelian. In light of Theorem 2.2 again, we deduce that a Sylow
$2$-subgroup of $G$ is dihedral of order $8$ and $M=Z_{2}\times Z_{2}$. Since
${\rm Aut}(M)\cong{\rm GL}_{2}(2)$, we have $H/L\cong Z_{3}$ and $HB/L\cong
S_{3}$. We finally conclude that $G/L\cong S_{4}$ and $|L|$ is odd. ∎

##  4\. Tidy $\\{p,q\\}$-groups

In this section, we classify the tidy $\\{p,q\\}$-groups. We first consider
the case where the Sylow $2$-subgroup is generalized quaternion and $O_{2}(G)$
has order $2$.

######  Lemma 4.1.

Let $G$ be a solvable tidy group. Suppose a Sylow $2$-subgroup $T$ of $G$ is
generalized quaternion. If $|O_{2}(G)|=2$, then $T$ is a quaternion group of
order $8$, $G$ has a normal $2$-complement, and $|F(G):C_{F(G)}(T)|$ is
divisible by at least two primes.

###### Proof.

The hypothesis that $|O_{2}(G)|=2$ implies that $Z=O_{2}(G)\leq Z(G)$. Since
$G$ is not abelian, we see that $Z(G)\neq F(G)$. Thus, $O_{2^{\prime}}(G)>1$.
In particular, there is an odd prime $p$ so that $O_{p}(G)>1$. Let
$p_{1},\dots,p_{n}$ be the odd prime divisors of $|F(G)|$ so that
$F(G)=Z\times F^{2^{\prime}}(G)$ where
$F^{2^{\prime}}(G)=O_{p_{1}}(G)\times\cdots\times O_{p_{n}}(G)$ is the Hall
$2$-complement of $F(G)$. Since $Z$ is central, we know that
$C_{G}(F^{2^{\prime}}(G))=C_{G}(F(G))\leq F(G)$, so
$C_{T}(F^{2^{\prime}}(G))\leq T\cap F(G)=Z$. Since $Z\leq
C_{T}(F^{2^{\prime}}(G))$, we deduce that $C_{T}(F^{2^{\prime}}(G))=Z$.
Observe that no quotient of $T/Z$ is generalized quaternion.

By Lemma 2.7, we see that every element of $T/C_{T}(O_{p_{i}}(G))$ acts
Frobeniusly on $O_{p_{i}}(G)$, and so, $T/C_{T}(O_{p_{i}}(G))$ is a Frobenius
complement. Since it is not generalized quaternion, we conclude that
$T/C_{T}(O_{p_{i}}(G))$ is cyclic for each value of $i$. This implies that
$T^{\prime}\leq C_{T}(O_{p_{i}}(G))$ and thus,
$T^{\prime}\leq\cap_{i=1}^{n}C_{T}(O_{p_{i}}(G))=C_{T}(F^{2^{\prime}}(G))=Z$.
This implies that $T$ has nilpotence class $2$. The only generalized
quaternion group with nilpotence class $2$ is the quaternion group of order
$8$, and thus, $T$ is the quaternion group. Notice that
$T^{\prime}<C_{T}(O_{p_{i}}(G))$, so we would have a contradiction if only one
prime divides $|F(G):C_{F(G)}(T)|$. Thus, at least two primes divide
$|F(G):C_{F(G)}(T)|$.

To complete the proof, we need to show that $G$ has a normal $2$-complement.
Let $H$ be a $2$-complement of $G$. If $C_{T}(O_{p_{i}}(G))<T$ for some $i$,
then $G/C_{G}(O_{p_{i}}(G))$ has a Sylow $2$-subgroup of order $2$. By a
corollary to the Burnside normal $p$-complement theorem (Corollary 5.14 of
[11]), this implies that $G/C_{G}(O_{p_{i}}(G))$ has a normal $2$-complement.
Hence, we have $HC_{G}(O_{p_{i}}(G))$ is normal in $G$ for each such $i$. On
the other hand, if $C_{T}(O_{p_{i}}(G))=T$, then $HC_{G}(O_{p_{i}}(G))=G$. We
deduce that $HC_{G}(O_{p_{i}}(G))$ is normal in $G$ for all $i$. This implies
that $\cap_{i=1}^{n}HC_{G}(O_{p_{i}}(G))=H(\cap_{i=1}^{n}C_{G}(O_{p_{i}}(G)))$
is a normal subgroup of $G$. Observe that $HZ\leq
H\cap_{i=1}^{n}C_{G}(O_{p_{i}}(G))=HC_{G}(F^{2^{\prime}}(G))\leq HF(G)=HZ$.
Thus, $HZ$ is normal in $G$. Since $Z$ is central, we see that $H$ is normal
and thus characteristic in $HZ$. We now obtain the desired conclusion that $H$
is normal in $G$. ∎

We next consider the centralizer $C_{H}(O_{2}(G))$ for a group $G$ where a
Sylow $2$-subgroup of $G$ is generalized quaternion group and $H$ is a
$2$-complement of $G$. Note the similarity to Theorem 2.1.

######  Lemma 4.2.

Let $G$ be a tidy group. Suppose that a Sylow $2$-subgroup of $G$ is a
generalized quaternion group. Let $H$ be a $2$-complement of $G$ and let
$C=C_{H}(O_{2}(G))$. Then $C$ is normal in $G$.

###### Proof.

Since $C_{G}(O_{2}(G))$ is normal in $G$, we see that $C$ is normal in $H$. It
follows that $C$ is a Hall $2$-complement of $C_{G}(O_{2}(G))$, and
$G/C_{G}(O_{2}(G))$ is isomorphic to a subgroup of ${\rm Aut}(O_{2}(G))$. We
know that $O_{2}(G)$ is either generalized quaternion or cyclic. If $O_{2}(G)$
is not the quaternion group, then ${\rm Aut}(O_{2}(G))$ is a $2$-group. Let
$T$ be a Sylow $2$-subgroup of $G$, and observe that $T\cap C_{G}(O_{2}(G))$
is a Sylow $2$-subgroup of $C_{G}(O_{2}(G))$. In particular, $O_{2}(G)$ is a
normal subgroup of $T$. Now, $T$ has a cyclic subgroup $D$ of index $2$. If
$O_{2}(G)$ is generalized quaternion, then $O_{2}(G)$ will contain elements
outside of $D$, and it will intersect $D$ in a subgroup of order at least $4$.
Observe that if $x\in T\setminus D$, then $C_{T}(x)=\langle x\rangle$, and if
$x\in D$, then $C_{T}(x)=D$. We deduce that if $O_{2}(G)$ is generalized
quaternion, then $C_{T}(O_{2}(G))=Z(T)$.

Since $O_{2}(G)$ is normal in $T$, if $O_{2}(G)$ is cyclic, then either
$O_{2}(G)\leq D$ or $C_{T}(O_{2}(G))=O_{2}(G)$. Hence, when $O_{2}(G)$ is
cyclic of order at least $4$, either $C_{T}(O_{2}(G))=D$ or
$C_{T}(O_{2}(G))=O_{2}(G)$. Thus, when $O_{2}(G)$ is generalized quaternion or
cyclic of order at least $4$, the Sylow $2$-subgroup of $C_{G}(O_{2}(G))$ is
cyclic. By Lemma 2.3, $C_{G}(O_{2}(G))$ has a normal $2$-complement. This
implies that $C$ is normal in $C_{G}(O_{2}(G))$ and hence characteristic in
$C_{G}(O_{2}(G))$. Since $C_{G}(O_{2}(G))$ is normal in $G$, we deduce that
$C$ is a normal subgroup of $G$. The remaining possibility is that $O_{2}(G)$
has order $2$. This implies that $C=H$. By Lemma 4.1, we conclude that $H$ is
normal in $G$, and the result is proved. ∎

In this next lemma, we classify the groups where a Sylow $2$-subgroup is
generalized quaternion and there is a central element of order $2$. This is a
key step in the general classification.

######  Lemma 4.3.

Let $G$ be a solvable, tidy group. Suppose a Sylow $2$-subgroup of $G$ is
generalized quaternion. If $x\in Z(G)$ has order $2$, then one of the
following occurs:

  1. (1)

$G$ has a normal $2$-complement.

  2. (2)

$O_{2}(G)$ is the quaternion group of order $8$ and if $H$ is a Hall
$\\{2,3\\}$-subgroup of $G$, then $H/O_{3}(H)$ is isomorphic to ${\rm
SL}_{2}(3)$ or $\widetilde{{\rm GL}_{2}(3)}$ and there is a Hall
$\\{2,3\\}$-complement $T$ so that $O_{3}(H)T$ is a normal subgroup of $G$.

###### Proof.

Since $G$ has a central element of order $2$, we must have that $O_{2}(G)>1$.
If $|O_{2}(G)|=2$, then by Lemma 4.1, we see that $G$ has a normal
$2$-complement. Suppose that $O_{2}(G)$ is cyclic or is generalized quaternion
of order at least $16$. We know that $C_{G}(O_{2}(G))$ is normal in $G$ and
$G/C_{G}(O_{2}(G))$ is isomorphic to a subgroup of ${\rm Aut}(O_{2}(G))$. This
implies that $G/C_{G}(O_{2}(G))$ is a $2$-group. By Lemmas 2.3 or 4.2,
$C_{G}(O_{2}(G))$ has a normal $2$-complement $C$. Observe that $C$ will be
characteristic in $G_{G}(O_{2}(G))$ and hence it is normal in $G$. Since
$|G:C_{G}(O_{2}(G))|$ is a $2$-power $C$ is a $2$-complement of $G$.

Thus, we are left with the case that $O_{2}(G)$ is the quaternion group of
order $8$. We know that $G/C_{G}(O_{2}(G))$ is isomorphic to a subgroup of
${\rm Aut}(O_{2}(G))\cong S_{4}$. Notice that the Sylow $2$-subgroup of
$C_{G}(O_{2}(G))$ will be $Z(O_{2}(G))$ which has order $2$, and

$\frac{O_{2}(G)C_{G}(O_{2}(G))}{C_{G}(O_{2}(G))}\cong Z_{2}\times Z_{2}.$

Thus, there are three possibilities here. In particular, $G/C_{G}(O_{2}(G))$
is isomorphic to (1) a Klein $4$-group, (2) $A_{4}$, or (3) $S_{4}$. We may
use Lemma 2.3 to see that $C_{G}(O_{2}(G))$ has a normal $2$-complement $C$.
In the first case, it is not difficult to see that $C$ is a normal
$2$-complement of $G$.

We assume that we are in cases (2) or (3). Let $H$ be a Hall
$\\{2,3\\}$-subgroup of $G$ and take $T$ to be a Hall $\\{2,3\\}$-complement
of $C$. Observe that $H\cap C$ will be a $3$-subgroup, will be normal in $H$
and will be a Sylow $3$-subgroup of $C$. Also,
$C_{H}(O_{2}(G))=Z(O_{2}(G))\times(H\cap C)$. We know that
$H/C_{H}(O_{2}(G))\cong G/C_{G}(O_{2}(G))$ is either $A_{4}$ or $S_{4}$. In
either case, we see that $H\cap C=O_{3}(H)$ and $C=O_{3}(H)T$. Since a Sylow
$2$-subgroup of $H$ is either the quaternion group of order $8$ or the
generalized quaternion group of order $16$, we see that $H/O_{3}(H)$ is
isomorphic to either ${\rm SL}_{2}(3)$ or $\widetilde{{\rm GL}_{2}(3)}$.
Therefore, we obtain conclusion (2). ∎

We also need to understand the centralizer when a Sylow subgroup is cyclic. We
obtain additional information in this case. We define the terms of the upper
central series as follows. Set $Z_{0}(G)=1$ and for $i\geq 1$, define
$Z_{i}(G)/Z_{i-1}(G)=Z(G/Z_{i-1}(G))$. It is easy to see that $Z_{1}(G)=Z(G)$
and $Z_{0}(G)\leq Z_{1}(G)\leq\dots$. Since $G$ is finite, there is an integer
$N$ so that $Z_{n}(G)=Z_{N}(G)$ for all $n\geq N$. We say that $Z_{N}(G)$ is
the hyper-center of $G$, and we denote it by $Z_{\infty}(G)$.

######  Lemma 4.4.

Let $G$ be a tidy group and let $p$ be a prime. Suppose that $G$ has a Sylow
$p$-subgroup $P$ that is cyclic and let $H$ be a Hall $p$-complement of $G$.
If $O_{p}(G)>1$ and $C=C_{H}(O_{p}(G))$, then $C$ is normal in $G$. In
addition, if $C<H$, then $G/C$ is a Frobenius group with Frobenius kernel
$P\cong PC/C$ and Frobenius complement $H/C$ that is cyclic of order dividing
$p-1$, $P\cap Z_{\infty}(G)=1$.

###### Proof.

Notice that $C_{G}(O_{p}(G))$ is normal in $G$, so $H\cap C_{G}(O_{p}(G))=C$
is a Hall $p$-complement of $C_{G}(O_{p}(G))$. There is an element $x$ so that
$O_{p}(G)=\langle x\rangle$. Then $C_{G}(x)=C_{G}(O_{p}(G))$. By Lemma 2.3,
$C_{G}(x)$ has a normal $p$-complement. This implies that $C$ is normal and
hence, characteristic in $C_{G}(O_{p}(G))$. Since $C_{G}(O_{p}(G))$ is normal
in $G$, we conclude that $C$ is normal in $G$.

Suppose $C<H$. Observe that $PC=C_{G}(O_{p}(G))$. Also, $H/C\cong
G/C_{G}(O_{p}(G))$ is isomorphic to a subgroup of ${\rm Aut}(O_{p}(G))$. Since
$O_{p}(G)$ is cyclic of prime power order and $|H:C|$ is coprime to $p$, we
conclude that $H/C$ is cyclic of order dividing $p-1$. (Recall that the
automorphism group of a cyclic $2$-group is a $2$-group and when $p$ is odd,
the automorphism group of a cyclic group of order $p^{a}$ is cyclic of order
$\phi(p^{a})=p^{a-1}(p-1)$.) It is not difficult to see that any nontrivial
automorphism of coprime order of a cyclic group of prime power order will be
fixed point-free. (This strongly uses the fact that we have prime power order;
it is not true if you do not assume this.) Thus, $O_{p}(G)H/C$ is a Frobenius
group with Frobenius kernel $O_{p}(G)C/C$ and Frobenius complement $H/C$. This
implies $Z(G)\cap O_{p}(G)=1$. Notice that $Z(G)\cap P$ is a normal
$p$-subgroup of $G$, and so, we deduce that $Z(G)\cap P\leq O_{p}(G)$ and so,
$Z(G)\cap P=1$. This implies that $Z_{\infty}(G)\cap P=1$.

Note that $|G:PC|=|H:C|=|H:O_{p}(G)C|$ divides $p-1$. Since the number of
Sylow $p$-subgroups of $G/C$ needs to be congruent to $1$ modulo $p$, we
deduce that $PC/C$ is normal in $G/C$. Hence, we have that $PC$ is normal in
$G$. By Fitting’s Lemma, we see that $H/C$ acts Frobeniusly on $PC/C$. ∎

We also make use of the following technical lemma about hypercenters.

######  Lemma 4.5.

Let $G$ be a tidy solvable group and let $p$ be a prime. Suppose that $G$ has
a Sylow $p$-subgroup $P$ and let $H$ be a Hall $p$-complement of $G$. If
$O_{p}(G)>1$ and $H\leq C_{G}(O_{p}(G))$, then $Z(G)\cap
P=Z(P)\cap(O_{p}(G))>1$. In particular, $O_{p}(G)=Z_{\infty}(G)\cap P$.

###### Proof.

Observe that $G=PH$. Since $O_{p}(G)$ is a nontrivial normal subgroup of $P$,
we have that $Z(P)\cap O_{p}(G)>1$. Observe that both $P$ and $H$ centralize
$Z(P)\cap O_{p}(G)$, so $Z(P)\cap O_{p}(G)\leq Z(G)\cap P$. On the other hand,
$Z(G)\cap P\leq P$, so $Z(G)\cap P\leq Z(P)$ and $Z(G)\cap P$ is a normal
$p$-subgroup of $G$, so $Z(G)\cap P\leq O_{p}(G)$. We obtain $Z(G)\cap P\leq
Z(P)\cap O_{p}(G)$. We deduce that $Z(G)\cap P=Z(P)\cap O_{p}(G)$.

We claim that $Z_{n}(G)\cap P=Z_{n}(P)\cap O_{p}(G)$ for $n\geq 1$. We have
proved this for $n=1$. Suppose we know it is true for $m\geq 1$, and we now
prove it is true for $m+1$. Since $H$ centralizes $O_{p}(G)$, it follows that
$HZ_{m}(G)$ will centralize $(Z_{m+1}(P)\cap O_{p}(G))Z_{m}(G)/Z_{m}(G)$. It
is obvious that $PZ_{m}(G)/Z_{m}(G)$ centralizes $(Z_{m+1}(P)\cap
O_{p}(G))Z_{m}(G)/Z_{m}(G)$. We have that $Z_{m+1}(P)\cap O_{p}(G)\leq
Z_{m+1}(G)\cap P$. On the other hand, it is easy to see that $Z_{m+1}(G)\cap
P$ is contained in $Z_{m+1}(P)$ and it is a $p$-group that is normal in $G$,
so we have proved the claim. Since $O_{p}(G)\leq Z_{n}(P)$ for some $n$, we
have $O_{p}(G)\leq Z_{n}(G)\leq Z_{\infty}(G)$ as desired. ∎

We now arrive at the main theorem of this section, the classification of tidy
$\\{p,q\\}$-groups.

######  Theorem 4.6.

Suppose $G$ is a $\\{p,q\\}$-group for distinct primes $p$ and $q$. Then $G$
is tidy if and only if $G$ has tidy Sylow $p$\- and Sylow $q$-subgroups and
one of the following occurs:

  1. (1)

$G$ is nilpotent.

  2. (2)

Up to relabeling $p$ and $q$, $Z_{\infty}(G)$ is a $q$-group and
$G/Z_{\infty}(G)$ is a Frobenius group whose Frobenius kernel is the Sylow
$p$-subgroup.

  3. (3)

$\\{p,q\\}=\\{2,3\\}$, $O_{2}(G)$ is a Klein $4$-group, $G/O_{3}(G)\cong
S_{4}$ and $G/O_{2}(G)$ is a Frobenius group whose Frobenius kernel is the
Sylow $3$-subgroup of $G/O_{2}(G)$ and whose Frobenius complement has order
$2$. Also, $Z(G)=1$.

  4. (4)

$\\{p,q\\}=\\{2,3\\}$, $O_{2}(G)$ is a Sylow $2$-subgroup of $G$ and is the
quaternion group of order $8$, $G/O_{3}(G)\cong{\rm SL}_{2}(3)$. Also,
$Z_{\infty}(G)=Z(O_{2}(G))\times O_{3}(G)$.

  5. (5)

$\\{p,q\\}=\\{2,3\\}$, $O_{2}(G)$ is the quaternion group of order $8$,
$G/O_{3}(G)\cong\widetilde{{\rm GL}_{2}(3)}$ and $G/O_{2}(G)$ is a Frobenius
group whose Frobenius kernel is the Sylow $3$-subgroup of $G/O_{2}(G)$ and
whose Frobenius complement has order $2$. Additionally,
$Z_{\infty}(G)=Z(G)=Z(O_{2}(G))$.

###### Proof.

We begin by assuming that $G$ is a tidy group. This implies that $G$ has tidy
Sylow $p$\- and Sylow $q$-subgroups $P$ and $Q$ respectively. If both
$P=O_{p}(G)$ and $Q=O_{q}(G)$, then $G=P\texttimes Q$ and $G$ is nilpotent and
we have (1). We now assume that $G$ is not nilpotent; so both $P>1$ and $Q>1$
and either $O_{p}(G)<P$ or $O_{q}(G)<Q$. Without loss of generality, we assume
$O_{q}(G)<Q$. Notice that if $O_{p}(G)=1$, then $O_{p}(G)<P$. Thus, we may
assume that $O_{p}(G)>1$ and $O_{q}(G)<Q$. Furthermore, if both $O_{p}(G)>1$
and $1<O_{q}(G)<G$ and $2\in\\{p,q\\}$, we set $p=2$.

Set $C=C_{Q}(O_{p}(G))$. By Theorem 2.2, we know that $P$ has exponent $p$, is
cyclic, is dihedral, or is generalized quaternion. By Lemmas 2.8, 2.10, 4.2,
and 4.4 $C$ is normal in $G$. This implies that $C=O_{q}(G)$. By our
assumption, $C<Q$. In particular, $G$ does not have a normal $p$-complement.
Hence, if $P$ is a dihedral group, then $p=2$. Using Lemma 2.9, we have
$O_{2}(G)=Z_{2}\times Z_{2}$ and in view of Lemma 2.10, $G/C\cong S_{4}$. This
implies that $q=3$ and $C=O_{3}(G)$. Observe that $C_{P}(O_{3}(G))=O_{2}(G)$.
Notice that $QO_{2}(G)$ is a normal subgroup of $G$. Obviously, $P/O_{2}(G)$
acts Frobeniusly on $Q/O_{3}(G)$, since $G/O_{2}(G)O_{3}(G)\cong S_{3}$. By
Lemma 2.7, we have that $P/O_{2}(G)$ acts Frobeniusly on $O_{3}(G)$. It
follows that $G/O_{2}(G)$ is a Frobenius group with Frobenius kernel
$QO_{2}(G)/O_{2}(G)$ and a Frobenius complement of order $2$. It is not
difficult to see that $Z(G)=1$. This yields (3).

Next, suppose $P$ is generalized quaternion, so again $p=2$. We are assuming
that $O_{2}(G)>1$. This implies that $O_{2}(G)$ is either cyclic or
generalized quaternion. In either case, $O_{2}(G)$ has a characteristic
subgroup of order $2$. This implies that there exists $x\in Z(G)$ with
$o(x)=2$. Since we are assuming that $G$ does not have a normal
$2$-complement, we may apply Lemma 4.3 to see that $O_{2}(G)$ is the
quaternion group of order $8$ and $q=3$ and $G/O_{3}(G)$ is isomorphic to
either ${\rm SL}_{2}(3)$ or $\widetilde{{\rm GL}_{2}(3)}$. Now, if
$G/O_{3}(G)$ is isomorphic to ${\rm SL}_{2}(3)$, then $O_{2}(G)$ will be a
Sylow $2$-subgroup of $G$. This implies that $P=C_{P}(O_{3}(G))$. By Lemma
4.5, we see that $Z_{\infty}(G)\cap Q=O_{3}(G)$. Because
$G/(Z(O_{2}(G))O_{3}(G))\cong A_{4}$, we deduce that $Z_{\infty}(G)\cap
O_{2}(G)=Z(O_{2}(G))$. This proves (4).

Suppose $G/O_{3}(G)$ is isomorphic to $\widetilde{{\rm GL}_{2}(3)}$. Observe
that $O_{2}(G)=C_{P}(O_{3}(G))$. Notice that $QO_{2}(G)$ is a normal subgroup
of $G$. Obviously, $P/O_{2}(G)$ acts Frobeniusly on $Q/O_{3}(G)$, since
$G/O_{2}(G)O_{3}(G)\cong S_{3}$. By Lemma 2.7, we have that $P/O_{2}(G)$ acts
Frobeniusly on $O_{3}(G)$. It follows that $G/O_{2}(G)$ is a Frobenius group
with Frobenius kernel $QO_{2}(G)/O_{2}(G)$ and a Frobenius complement of order
$2$. This implies that $Z_{\infty}(G)\cap Q=1$. Since
$G/(Z(O_{2}(G))O_{3}(G))\cong S_{4}$, we deduce that $Z_{\infty}(G)\cap
P=Z(O_{2}(G))$. This proves (5).

We now assume $P$ is cyclic or has exponent $p$. By Lemma 4.4 or 2.8, we see
that $QO_{p}(G)/O_{q}(G)$ is a Frobenius group with Frobenius kernel
$PO_{q}(G)/O_{q}(G)$. Also, $P\cap Z_{\infty}(G)=1$. Observe that
$P=C_{P}(O_{q}(G))$. By Lemma 4.5, we have $O_{q}(G)=Z_{\infty}(G)\cap Q$. We
conclude that $Z_{\infty}(G)=O_{q}(G)$. If $G=O_{p}(G)Q$, then we obtain (2).
Thus, we may assume that $O_{p}(G)Q<G$, and this implies that $O_{p}(G)<P$.
Let $N$ be a normal subgroup so that $O_{p}(G)O_{q}(G)<N$ and
$N/O_{p}(G)O_{q}(G)$ is a chief factor for $G$. Thus, $O_{p}(G)O_{q}(G)$ is
either a $p$-group or a $q$-group.

Suppose first that $N/(O_{p}(G)O_{q}(G))$ is a $q$-group. (Note that we may
have $O_{q}(G)=1$.) Since $QO_{p}(G)/O_{q}(G)$ is a Frobenius group, we have
that $N/O_{q}(G)$ is a Frobenius group with $O_{p}(G)O_{q}(G)$ as its
Frobenius kernel. By the Frattini argument, we have $G=NN_{G}(N\cap
Q)=O_{p}(G)(N\cap Q)N_{G}(N\cap Q)=O_{p}(G)N_{G}(N\cap Q)$. It follows that
$P=O_{p}(G)N_{P}(N\cap Q)$. Since $(N\cap Q)/O_{q}(G)$ acts Frobeniusly on
$O_{p}(G)$, we see that $O_{p}(G)\cap N_{P}(N\cap Q)=1$. Hence, we can find
$1\neq x\in N_{P}(N\cap Q)$. Suppose there exists $y\in N\cap Q$ that is
centralized by $x$. Observe that $1<Z(P)\cap O_{p}(G)\leq C_{G}(x)$. By Lemma
2.5 or 2.3, we know that $C_{G}(x)$ has a normal $p$-complement. This implies
that $y$ is in a normal subgroup of $C_{G}(x)$ that is disjoint from $Z(P)\cap
O_{p}(G)$. It follows that $y$ centralizes $Z(P)\cap O_{p}(G)$. Since $(N\cap
Q)/O_{q}(G)$ is acting Frobeniusly on $O_{p}(G)$, we deduce that $y\in
O_{q}(G)$. We now see that $N_{P}(N\cap Q)$ is acting Frobeniusly on $(N\cap
Q)/O_{q}(G)$. Since $N/(O_{p}(G)O_{q}(G))\cong(N\cap Q)/O_{q}(G)$, we see that
$NP/(O_{p}(G)O_{q}(G))$ is a Frobenius group with Frobenius kernel
$N/(O_{p}(G)O_{q}(G))$. Hence, $NP/O_{q}(G)$ is a $2$-Frobenius group. Observe
that $F(NP)=O_{p}(G)\times O_{q}(G)$. Thus, Lemma 3.3 applies and implies that
$p=2$ and $NP/O_{p}(G)\cong S_{4}$. This however contradicts the fact that a
Sylow $p$-subgroup is cyclic or has exponent $p$. This case cannot occur.

We have that $N/(O_{p}(G)O_{q}(G))$ is a $p$-group. This implies that
$O_{q}(G)>1$ since $N>O_{p}(G)O_{q}(G)$. Notice that $O_{p}(G)<P$ and
$1<O_{q}(G)<Q$ implies that if $2\in\\{p,q\\}$, then $p=2$. Observe that
$N\cap P$ is not normal in $G$, so $N\cap P$ cannot centralize $O_{q}(G)$.
Using Lemma 2.7, we see that $(N\cap P)/O_{p}(G)$ acts Frobeniusly on
$O_{q}(G)$. (Since $q\neq 2$, we have that conclusion (3) of Lemma 2.7 does
not occur.) Since $q\neq 2$, we know from Theorem 2.2 that $Q$ is either
cyclic or exponent $q$. If $Q$ is cyclic, then Lemma 4.4 would apply and
$O_{q}(G)=Q$ which is a contradiction to the choice of $p$ and $q$. Hence $Q$
has exponent $q$. Now, we can reverse the roles of $p$ and $q$, and the same
argument as in the last paragraph applies to yield a contradiction. This
completes this direction.

To see the converse, observe that a nilpotent group where all the Sylow
subgroups are tidy is tidy, so we have the result if (1) holds. If (3), (4),
or (5) hold, then $G$ is tidy by Theorem 3.2. If (2) holds, then we have that
$G$ is tidy by Theorem 3.1 where $N=Z_{\infty}(G)$ and $H=Q$ is nilpotent. ∎

##  5\. Solvable tidy groups

In this section, we work to prove some consequences of our characterization of
solvable tidy groups.

We now work to prove that the quotients of finite solvable tidy groups are
tidy. We will see that the proof relies on the fact that a finite solvable
group is tidy if and only if its Hall $\\{p,q\\}$-subgroups are tidy for all
primes $p$ and $q$ and on the classification of tidy $\\{p,q\\}$-groups. At
this time, it is an open question as to whether quotients of finite
nonsolvable tidy groups are tidy.

We begin with the observation using Theorem 2.2 that any quotient of a tidy
$p$-group will be tidy where $p$ is a prime. Next, we use Theorem 4.6 to
determine the tidy $\\{p,q\\}$-group where $p$ and $q$ are distinct primes.
This will be the key to the next lemma.

######  Lemma 5.1.

Let $G$ be a tidy $\\{p,q\\}$-group for distinct primes $p$ and $q$. If $N$ is
a normal subgroup of $G$, then $G/N$ is a tidy group.

###### Proof.

We know that all subgroups of tidy groups are tidy, so the Sylow $p$\- and
Sylow $q$-subgroups of $G$ are tidy. Hence, if $G/N$ is a $p$-group or a
$q$-group, then it is a quotient of a tidy $p$-group or $q$-group, and we have
seen that quotients of tidy $p$-groups are tidy. Thus, we may assume that
$G/N$ is not a $p$-group or a $q$-group. We now consider the possibilities
from Theorem 4.6. If $G$ is nilpotent, then $G/N$ is nilpotent, and the Sylow
$p$\- and Sylow $q$-subgroups are quotients of tidy groups, so they are tidy.
Thus, $G/N$ is tidy. Suppose $Z_{\infty}(G)$ is a $q$-group and
$G/Z_{\infty}(G)$ is a Frobenius group. Let $F=F(G)$ and observe that
$F=P\times Z_{\infty}(G)$ where $P$ is the Sylow subgroup. Then
$FN/N=PN/N\times Z_{\infty}(G)N/N$ and $G/(Z_{\infty}(G)N)$ will be a
Frobenius group with Frobenius kernel $FN/N$. By Theorem 3.1, we see that
$G/N$ is tidy. Finally, suppose $G$ satisfies (3), (4), or (5) of Theorem 4.6.
Observe that the possible quotients of $G$ that are not $p$-groups or
$q$-groups are $S_{3}$, $A_{4}$, or a group that satisfies the hypotheses of
Theorem 3.2. Certainly, one can use Proposition 2.5 of [1] to see that $S_{3}$
and $A_{4}$ are tidy, and Theorem 3.2 shows that remaining possible quotients
are tidy. ∎

We now prove that quotients of solvable tidy groups are tidy.

###### Proof of Theorem 2.

Let $\pi$ be the set of primes that divide $|G|$, and let $\rho\subseteq\pi$
have size $2$. Let $H$ be a Hall $\rho$-subgroup of $G$. Then $HN/N$ is a Hall
$\rho$-subgroup of $G/N$. Since $G$ is tidy, we see that $H$ is tidy. By Lemma
5.1, we see that $HN/N$ is tidy. Thus, for every two element subset $\rho$ of
$\pi$, we see that $G/N$ has a tidy Hall $\rho$-subgroup. By Theorem 1.1 of
[6], we see that $G/N$ is tidy. ∎

We next work to prove that the Fitting height of a tidy solvable group is at
most $4$. We first prove a theorem that classifies the groups that arise
modulo the centralizer of $O_{p}(G)$. This is the key step in bounding the
Fitting height.

######  Theorem 5.2.

Let $G$ be a solvable tidy group, let $p$ be a prime, let $H$ be a Hall
$p$-complement of $G$ and let $C=C_{H}(O_{p}(G))$. If $O_{p}(G)>1$, then $C$
is a normal subgroup of $G$ and one of the following occurs:

  1. (1)

$G/C$ is a $p$-group.

  2. (2)

$G/C$ is a Frobenius group whose Frobenius kernel is $O_{p}(G)C/C$.

  3. (3)

$p=2$, $O_{2}(G)$ is the quaternion group of order $8$, and $G/C$ is
isomorphic to ${\rm SL}_{2}(3)$ or $\widetilde{{\rm GL}_{2}(3)}$.

  4. (4)

$p=2$, $O_{2}(G)$ is the Klein $4$-group and $G/C$ is isomorphic to $S_{4}$.

  5. (5)

The following occur:

    1. (a)

$p=3$.

    2. (b)

$G/C$ is a Frobenius group whose Frobenius kernel is a Sylow $3$-subgroup and
whose Frobenius complement has order $2$.

    3. (c)

$O_{2}(G)$ is either a Klein $4$-group or is a quaternion group of order $8$.

    4. (d)

$O_{3}(G)$ has index $3$ in a Sylow $3$-subgroup of $G$.

    5. (e)

$G/O_{3}(G)C\cong S_{3}$.

###### Proof.

We know by Theorem 2.2 that a Sylow $p$-subgroup of $G$ is one of the
following: exponent $p$, cyclic, dihedral, or generalized quaternion. Thus,
applying Lemmas 2.8, 2.10, 4.2 and 4.4,we see that $C$ is normal in $G$.
Suppose first $p=2$ and that a Sylow $2$-subgroup is a generalized quaternion
group. Since $O_{2}(G)>1$, this implies that $G$ has a central element of
order $2$. We may apply Lemma 4.3 to see that either conclusion (1) or (3)
occurs. Next, suppose that $p=2$ and $G$ has a dihedral Sylow $2$-subgroup. By
Lemma 2.10, we see that either conclusion (1) or (4) holds.

Next, suppose that a Sylow $p$-subgroup is cyclic or has exponent $p$. Let $q$
be a prime distinct from $p$ that divides $|G:C|$. If either $p\neq 3$ or
$q\neq 2$ when $p=3$, we see from Theorem 4.6 that $PQC/C$ is either nilpotent
or a Frobenius group where $P$ is normalized by $Q$. Notice that if $PQC/C$ is
nilpotent, then $Q$ would centalize $O_{p}(G)$ and this would imply that
$Q\leq C$ contradicting $q$ divides $|G:C|$. Hence, we have that $PQC/C$ is a
Frobenius group. Since all the Sylow subgroups of $G/C$ normalize $P$, we
conclude that $P=O_{p}(G)$. Notice that if $1\neq x\in P$ centralizes a
nontrivial element of $H/C$, then it will centralize a nontrivial element of
prime power order, and this is a contradiction. Hence, we have $G/C$ is a
Frobenius group with Frobenius kernel $P=O_{p}(G)\cong O_{p}(G)C/C$. This
yields (2).

We are left with case that $p=3$ and $2$ divides $|G:C|$. Let $T$ be a Sylow
$2$-subgoup of $G$. If $PTC/C$ is nilpotent or a Frobenius group with
Frobenius kernel $P$, then we can use the argument in the previous paragraph
again to see that we have (2) again. This handles conclusions (1) and (2) of
Theorem 4.6. We now suppose that $PTC/C$ satisfies (3), (4), or (5) from
Theorem 4.6. In all three of these cases, we see that $|P:O_{3}(G)|=3$. We
have seen that if $q$ is a prime divisor of $|G:C|$ other than $2$ or $3$,
then $QC/C$ will act Frobeniusly on $P$. This would imply that it acts
Frobeniusly on $P/O_{3}(G)$ which is a contradiction since $|P:O_{3}(G)|=3$.
Hence, no prime other than $2$ or $3$ divides $|G:C|$. In conclusion (4) of
Theorem 4.6, we see that $G/C$ is a $3$-group, and we have conclusion (1). In
conclusions (3) or (5) of Theorem 4.6, we obtain conclusion (5). This proves
the theorem. ∎

We now are able to bound the Fitting height. We also bound the derived length
of $G/F(G)$. Since one can find Frobenius kernels of exponent $p$ and
arbitrarily large derived length, we are not able to bound the derived length
of $G$. In particular, we prove Theorem 1.

###### Proof of Theorem 1.

Let $p_{1},\dots,p_{n}$ be the prime divisors of $F(G)$, so that
$F(G)=O_{p_{1}}(G)\times\cdots\times O_{p_{n}}(G)$. Let $H_{i}$ be a Hall
$p_{i}$-complement of $G$ for each $i=1,\dots,n$ and let
$C_{i}=C_{H_{i}}(O_{p_{i}}(G))$. By Theorem 5.2, we know that $G/C_{i}$ is one
of: (1) a $p_{i}$-group; (2) a Frobenius group with Frobenius complement
$O_{p_{i}}(G)C_{i}/C_{i}$; (3) $p_{i}=2$ and $G/C_{i}$ is isomorphic to
$S_{4}$, ${\rm SL}_{2}(3)$, or $\widetilde{{\rm GL}_{2}(3)}$; or (4)
$p_{i}=3$, $G/C_{i}$ is a Frobenius group whose Frobenius kernel is the Sylow
$3$-subgroup of $G/C_{i}$, a Frobenius complement is cyclic of order $2$, and
$G/O_{3}(G)C_{i}\cong S_{3}$. Note that if $G/(O_{p_{i}}(G)C_{i})$ is
isomorphic to a Frobenius complement, then it has normal subgroup that is
metacyclic and whose quotient is isomorphic to a subgroup of $S_{4}$. It is
not hard to show that $G/(O_{p_{i}}(G)C_{i})$ has Fitting height at most $3$
and derived length at most $4$ in all cases. If $|G|$ is odd, then
$G/(O_{p_{i}}(G)C_{i})$ is either cyclic or metacyclic; so it has Fitting
height and derived length at most $2$.

Take $N=\cap_{i=1}^{n}O_{p_{i}}(G)C_{i}$. It follows that $G/N$ has Fitting
height at most $3$ and derived length at most $4$ and if $|G|$ is odd, then
$G/N$ has Fitting height and derived length at most $2$. Note that if $n=1$,
then $C_{G}(O_{p_{1}}(G))=C_{G}(F(G))\leq F(G)$, and so, $C_{1}=1$. This
implies that $N=O_{p_{1}}(G)C_{1}=O_{p_{1}}(G)=F(G)$. Suppose that $n\geq 2$.
Consider an element $x\in N$. We can write $x=x_{1}\cdots x_{n}x^{\prime}$
where $x_{1},\dots,x_{n},x^{\prime}$ are powers of $x$ and each $x_{i}$ has
$p_{i}$-power order and $x^{\prime}$ has
$\\{p_{1},\dots,p_{n}\\}^{\prime}$-order. Since $x\in O_{p_{i}}(G)C_{i}$, we
see that $x_{i}\in O_{p_{i}}(G)$ for each $i$. Also, $x^{\prime}\in C_{i}$ for
each $i$. This implies that $x^{\prime}$ centralizes $O_{p_{i}}(G)$ for each
$i$. We deduce that $x^{\prime}$ centralizes $F(G)$, and so, $x^{\prime}\in
C_{G}(F(G))\leq F(G)$. Since the order of $x^{\prime}$ is coprime to $|F(G)|$,
we see that $x^{\prime}=1$. Now, $x=x_{1}\cdots x_{n}\in
O_{p_{1}}(G)\times\cdots\times O_{p_{n}}(G)=F(G)$. We conclude that $N\leq
F(G)$. We have now proved the result. ∎

## References

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  * [6] N. F. Beike, R. Carleton, D. G. Costanzo, C. Heath, M. L. Lewis, K. Lu, and J. D. Pearce, Finite solvable tidy groups are determined by Hall subgroups with two primes, to appear in Bull. Aust. Math. Soc. DOI: https://doi.org/10.1017/S0004972723000710
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         https://scholar.rose-hulman.edu/math_mstr/131


[1]I. M. Carvalho

[1]Institute of Chemistry, São Paulo State University, Francisco Degni 55,
Araraquara, 14800-090, São Paulo, Brazil

# Quantum phase transitions in one-dimensional nanostructures: a comparison
between DFT and DMRG methodologies

T. Pauletti<EMAIL_ADDRESS>M. Sanino<EMAIL_ADDRESS>L.
Gimenes<EMAIL_ADDRESS><EMAIL_ADDRESS>V. V. França
<EMAIL_ADDRESS>*

###### Abstract

Context: In the realm of quantum chemistry, the accurate prediction of
electronic structure and properties of nanostructures remains a formidable
challenge. Density Functional Theory (DFT) and Density Matrix Renormalization
Group (DMRG) have emerged as two powerful computational methods for addressing
electronic correlation effects in diverse molecular systems. We compare
ground-state energies ($e_{0}$), density profiles ($n$) and average
entanglement entropies ($\bar{S}$) in metals, insulators and at the transition
from metal to insulator, in homogeneous, superlattices and harmonically
confined chains described by the fermionic one-dimensional Hubbard model.
While for the homogeneous systems there is a clear hierarchy between the
deviations, $D\%(\bar{S})<D\%(e_{0})<\bar{D}\%(n)$, and all the deviations
decrease with the chain size; for superlattices and harmonical confinement the
relation among the deviations is less trivial and strongly dependent on the
superlattice structure and the confinement strength considered. For the
superlattices, in general increasing the number of impurities in the unit cell
represents less precision on the DFT calculations. For the confined chains,
DFT performs better for metallic phases, while the highest deviations appear
for the Mott and band-insulator phases. This work provides a comprehensive
comparative analysis of these methodologies, shedding light on their
respective strengths, limitations, and applications.

Methods: The DFT calculations were performed using the standard Kohn-Sham
scheme within the BALDA approach. It integrated the numerical Bethe-Ansatz
(BA) solution of the Hubbard model as the homogeneous density functional
within a local-density approximation (LDA) for the exchange-correlation
energy. The DMRG algorithms were implemented using the ITensor library, which
is based on the Matrix Product States (MPS) ansatz. The calculations were
performed until the energy reaches convergence of at least $10^{-8}$.

###### keywords:

Density Functional Theory, Density Matrix Renormalization Group, Hubbard
Model, Quantum Phase Transitions, Entanglement.

##  1 Introduction

Reliable predictions of nanostructures properties are essential for the
development of advanced quantum technologies. Density Functional Theory (DFT)
[1, 2] and Density Matrix Renormalization Group (DMRG) [3] methods stand out
prominently in this pursuit. DMRG has demonstrated remarkable success in
capturing strong correlation effects by systematically optimizing the
wavefunction in a reduced Hilbert space, but gets computationally expensive
with the size of the system.

On the other hand, DFT offers a cost-effective approach to study large systems
and is widely used as an electronic structure method. However, the inherent
approximations in exchange-correlation functionals can lead to inaccuracies,
particularly in strongly correlated systems. Despite of that Kohn-Sham DFT [4,
5] has been demonstrated to be reliable to describe for example i) transport
properties in the weakly coupled repulsive regime [6], ii) phases in cold
trapped atoms [7, 8], iii) charge gap [9], and iv) electrical response
properties of homogeneous metals [10].

In this work, we provide a detailed comparative assessment of DFT and DMRG in
nanostructures described by the one-dimensional Hubbard model. We explore
homogeneous, superlattices and harmonically confined chains, addressing key
properties such as ground-state energies ($e_{0}$), density profiles ($n$) and
average entanglement entropies ($\bar{S}$), the latter being commonly used to
detect and characterize quantum phase transitions. While for the homogeneous
systems one finds a clear hierarchy between the deviations,
$D\%(\bar{S})<D\%(e_{0})<\bar{D}\%(n)$, for superlattices and harmonically
confined chains the relation among deviations is non-trivial and dependent on
the superlattice structure and the confinement strength adopted. For the
superlattices, in general increasing the number of impurities in the unit cell
makes DFT calculations less precise. For the confined systems, DFT performs
better for metallic phases, while the highest deviations appear for the Mott
and band-insulator phases.

##  2 Model and Methods

We consider one-dimensional (1D) systems described by the fermionic Hubbard
Hamiltonian,

$\displaystyle\hat{H}=-t\sum_{i,\sigma}(\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i+1,\sigma}+\hat{c}_{i,\sigma}\hat{c}^{\dagger}_{i+1,\sigma})+U\sum_{i}\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}+\sum_{i,+-\sigma}V_{i}\hat{n}_{i,\sigma},$
(1)

where $\hat{c}^{(\dagger)}_{i,\sigma}$ annihilates (creates) an electron with
spin $\sigma=\uparrow,\downarrow$ at site $i$, and
$\hat{n}_{i,\sigma}=\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma}$ are number
operators. The model considers nearest-neighbour hopping, on-site interactions
and local external potential terms with coefficients $t$, $U$, and $V_{i}$,
respectively. The ground-state is calculated by fixing the total number of
electrons $N$ in a chain of size $L$ and the total null magnetization, such
that
$\sum_{i}\left\langle\hat{n}_{i,\uparrow}\right\rangle=\sum_{i}\left\langle\hat{n}_{i,\downarrow}\right\rangle=N/2$.
We consider different average densities $n=N/L$, restricted to $0\leq n\leq 2$
(single energy band) and impose open boundary conditions at the chain ends.
Throughout this work, we set $t=1$, which defines the unit of energy.

At the limit $V_{i}=0$ the Hamiltonian of Eq. (1) reduces to the standard
Hubbard model. In this case, for $n<1$ the ground-state represents a metallic
phase, while for the half-filling ($n=1$) the system consists of a Mott
insulating phase for arbitrarily small $U>0$ [11]. When $V_{i}$ is
periodically modulated, thus simulating superlattices (as in Fig. 1b), the
potential may induce a quantum phase transition from metal to insulating [12,
13, 14]. Similarly the Mott metal-insulator transition may be induced by the
harmonical potential $V_{i}=k(i-L/2)^{2}$ (as in Fig. 1c), widely used in
state-of-the-art cold atoms experiments.

Figure 1: Schematic setup of (a) a finite homogeneous lattice with $V_{i}=0$
(blue circles), (b) superlattices with distinct modulations of the periodic
$V_{i}\neq 0$ (red circles), for a fixed unit cell of 9 sites ($2:7$, $3:6$
and $4:5$) and (c) a chain under a harmonic confinement centered at $L/2$,
$V_{i}=k{(i-L/2)}^{2}$.

In this paper, we address the performance of DFT $-$ when compared to
numerically exact DMRG calculations $-$ in describing the properties of the
Hubbard model for all the three 1D systems: homogenous, superlattices and
harmonically confined chains, as schematically illustrated in Figure 1.

We focus on three main quantities: the per-site ground-state energy $e_{0}$,
the density profile $\\{n_{i}\\}$ and its respective nonlocal correlations,
quantified by the average single-site entanglement

$\displaystyle\bar{S}=\frac{1}{L}\sum_{i}^{L}S_{i},$ (2)

with

$\displaystyle S_{i}=-{\rm Tr}[{\rho}_{i}{\rm
log}_{2}{\rho}_{i}]=-\sum_{k}{\rm w}_{i,k}{\rm log}_{2}{\rm w}_{i,k},$ (3)

where ${\rm w}{{}_{i,k}}$ are the eigenvalues of the $i$-site reduced density
matrix ${\rho}_{i}={\rm Tr}_{L-1}[\rho_{GS}]$ calculated by tracing out the
remaining $L-1$ sites from the ground-state density matrix $\rho_{GS}$. This
type of entanglement has been proved to be relevant in analyzing the
localization and itinerancy of the indistinguishable particles [15, 16, 17].
In the site-occupation basis the reduced Hilbert space has dimension $d=4$,
and thus ${\rm w}_{i,k}$ are the occupation probabilities with
$k=\uparrow,\downarrow,2,0$, such that $\sum_{k}{\rm w}_{i,k}=1$. The set
$\\{{\rm w}_{i,k}\\}$ can be calculated by firstly computing the paired
probability ${\rm
w}_{i,2}=\langle\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}\rangle=\partial
e_{0}/\partial U$. From this term one obtain the remaining probabilities: the
unpaired, ${\rm w}_{i,\uparrow}={\rm
w}_{i,\downarrow}=\left\langle\hat{n}_{i}\right\rangle/2-{\rm w}_{i,2}$, and
the empty probability ${\rm w}_{i,0}=1-{\rm w}_{i,\uparrow}-{\rm
w}_{i,\downarrow}-{\rm w}_{i,2}$. The DMRG algorithms were implemented using
the ITensor library [18]. This library is based on the Matrix Product States
(MPS) ansatz. In this context, up to $800$ sweeps were utilized to reach an
energy convergence of at least $10^{-8}$. The precision of the MPS
representations was essentially controlled by configuring the bond link, which
its maximum is setting to approximately $3000$.

Via DFT, the per-site ground-state energy $e_{0}$ and the density profile
$\\{n_{i}\\}$ are obtained via standard Kohn-Sham scheme [4, 5] within BALDA
approach: in which the numerical Bethe-Ansatz (BA) solution of the Hubbard
model is considered as the homogeneous density functional within a local-
density approximation (LDA) for the exchange-correlation energy (for a review
see [19]).

Now for the entanglement, which requires the energy derivative with respect to
$U$, ${\rm w}_{2}=\partial e_{0}/\partial U$, to avoid errors related to the
numerical derivative of the BALDA approach described above, we adopt instead
the analytical derivative of the FVC parameterization for the energy [20],
which (for non-magnetized systems) is given by

$\displaystyle{\rm
w}_{2}^{FVC}(n,U)=\frac{2n}{\beta(n,U)}\frac{\partial\beta(n,U)}{\partial
U}\cos\left(\frac{\pi
n}{\beta(n,U)}\right)-\frac{2}{\pi}\frac{\partial\beta(n,U)}{\partial
U}\sin\left(\frac{\pi n}{\beta(n,U)}\right),$ (4)

where $\beta(n,U)=\beta(U)^{\sqrt[3]{U}/8}$ and the function $\beta(U)$ is
determined from

$\frac{\beta\left(U\right)}{\pi}\sin\left(\frac{\pi}{\beta\left(U\right)}\right)=2\int_{0}^{\infty}{\rm
d}x\frac{J_{0}\left(x\right)J_{1}\left(x\right)}{x\left(1+e^{Ux/2}\right)},$
(5)

with $J_{0}$ and $J_{1}$ the zero and first order Bessel functions,
respectively. This analytical parametrization becomes exact by construction
for i) $U\rightarrow 0$ (for which $\beta=2$), ii) $U\rightarrow\infty$
($\beta=1$), and iii) $n=1$ and any $U$ ($0\leqslant\beta\leqslant 1$). For
other $(n,U)$ regimes the FVC parametrization provides a reasonable
approximation to the full Beth-Ansatz [21, 22] solution.

Figure 2: DMRG-DFT deviation for the ground-state energy, Eq.(7), as a
function of the chain length for a finite homogeneous system for low density
(a), intermediate density (b), and at half-filling (c), for several strengths
of interaction $U$.  Figure 3: Mean percentage deviation of the local
densities, Eq.(9), as a function of the chain length for a homogeneous system
at (a) $n=0.2$, (b) $n=0.6$, and (c) $n=1$ for several values of $U$.

We thus adopt Eq.(4) as approximation to the homogeneous chain and hence
perform a LDA to obtain the double occupancy for the inhomogeneous systems:

${\rm w}_{i,2}\approx{\rm w}_{i,2}^{LDA}={\rm w}_{2}^{FVC}(n,U)|_{n\rightarrow
n_{i}},$ (6)

by using the density profile $\\{n_{i}\\}$ obtained via BALDA.

The percentage deviations between DFT and DMRG calculations are then, for the
energy

$D\%(e_{0})=\left|\frac{e_{0}^{\rm DMRG}-e_{0}^{\rm DFT}}{e_{0}^{\rm
DMRG}}\right|\times 100,$ (7)

for the average single-site entanglement,

${D}\%(\bar{S})=\left|\frac{\bar{S}^{\rm DMRG}-\bar{S}^{\rm DFT}}{\bar{S}^{\rm
DMRG}}\right|\times 100,$ (8)

while for the density profiles we quantify a mean percentage deviation,
defined as:

$\bar{D}\%(n)=\frac{1}{L}\sum_{i}^{L}\left|\frac{n_{i}^{\rm DMRG}-n_{i}^{\rm
DFT}}{n_{i}^{\rm DMRG}}\right|\times 100.$ (9)

##  3 Results and Discussion

###  3.1 DFT x DMRG: energy, density profile and entanglement

First, we analyze the DFT performance in reproducing the DMRG results for
finite chains. In Figure 2 we present the percentage deviation for the per-
site ground-state energy. We see that in general DFT performs better for low
densities and weak interactions, in both cases due to less influence of the
electron-electron correlations. We find that the error monotonically decays by
increasing the chain length $L$ for all $n,U$, since our DFT-LDA approach
becomes exact in the limit of $L\rightarrow\infty$.

Now considering the DFT performance for the local densities $\\{n_{i}\\}$, we
show in Figure 3 the mean deviation defined in Eq.(9). For the Mott insulator
regime, $n=1$ and any $U>0$, DFT is very precise with average deviations
smaller than $0.004\%$ due to the suppression of the Friedel-type oscillations
[23, 24]. For the metallic regime, $n<1$, the open ends of the chain act as
effective impurities, inducing Friedel oscillations in the density
distribution, which are not completely described by DFT, as can be confirmed
by Figure 4. Since the Friedel oscillations are more pronounced for low
densities, the deviations is greater for $n=0.2$ (Fig. 3a). In contrast, the
smallest deviation for the energy is precisely for the case of $n=0.2$ (Fig.
2a). We attribute this to error cancellations: while Eq.(9) sum up the local
errors, the under and over DFT estimates for the density profile leads to
error cancellation for the total energy. Nevertheless, for chain sizes
$L\gtrsim 60$ DFT density profiles are also reliable, with mean deviations
smaller than $2\%$.

Figure 4: Density profile for the homogeneous system with $L=60$ sites, for
several interactions $U$ and for distinct average filling factors: (a)
$n=0.2$, (b) $n=0.6$, and (c) $n=1$.

For the average single-site entanglement, in Figure 5, one finds a good
performance of DFT calculations even for the worse regimes of parameters (for
$L=10$, $n=1$, and $U=4$): the maximum deviation observed is $1.73\%$. This
very good performance of DFT for the average entanglement also comes from
error cancellations: the von Neumann entropy is a density functional [15, 16],
thus it reflects the DFT difficulty of reproducing the Friedel oscillations by
under and overestimating the single-site entanglement along the chain, hence
leading to a good average. As $\bar{S}$ has been widely used to detect and
characterize quantum phase transitions [17, 25, 26, 27, 28, 29, 30, 31, 32],
our results then show that DFT represents a reliable and powerful method to be
used in Hubbard chains, as we explore below in inhomogeneous chains.

Figure 5: Average entanglement $\bar{S}$ as a function of the chain length of
homogeneous systems for several interaction regimes and different densities:
$n=0.2$ (a), $n=0.6$ (b), and $n=1$ (c).

###  3.2 DFT x DMRG: detecting quantum phase transitions

We focus on the performance of DFT calculations for identifying the Mott
metal-insulator transition in superlattices and harmonically confined chains.
Starting with the superlattices (with unit cell SL $2:7$), Figure 6 shows that
DFT is considerably precise in reproducing the ground-state energy: deviations
increase in general by increasing the interaction $U$ and the superlattice
potential $V$, but are smaller than $\sim 3\%$ for all the regimes of
parameters considered.

Figure 6:  DMRG-DFT deviation for the ground-state energy of the superlattice
$2:7$ as a function of interaction $U$, for several periodic potential
strengths: (a) $V=0$, (b) $V=-2$, (c) $V=-4$ and (d) $V=-10$.

Accordingly, Figure 7 reveals that also the density profile in the
superlattice is reasonably well reproduced by DFT. For non-interacting
systems, Fig. 7a, the density concentrates within the impurity sites, since
the modulated potential is attractive, $V<0$. For $U=0$ and low average
densities ($n=0.2$ and $n=0.4$), the particles essentially reside in the
impurity sites, emptying the non-impurity ones. For higher average densities
($n=0.8$ and $n=1$), the impurity sites reach maximum occupation ($n_{i}=2$),
thus the remaining particles are distributed also in the non-impurity sites.

Figure 7: Density profile for a superlattice $2:7$ with $V=-4$, $L=90$ sites,
for several densities and distinct interactions: (a) $U=0$, (b) $U=2$, (c)
$U=4$ and (d) $U=10$.

When turning the interaction on, Figs. 7b$-$d, we find that the non-impurity
sites start to be filled with particles even for the low density regime
$n=0.4$, once now the competition between $V<0$ and $U>0$ disfavors the
maximum occupation at the impurity sites. Thus depending on the three
parameters $V,n,U$ the interaction can be strong enough to avoid $n_{i}>1$ in
the impurity sites, thus characterizing a local Mott insulator phase [13, 12].
This is seen already for $U\leq 2$ at $n=0.2$, while at $n=0.4$ this appears
for $U\leq 4$ and at $n=0.8$ and $n=1$ only for $U\leq 10$, for this fixed
$V=-4$.

Figure 8: Average entanglement for a superlattice as a function of the on-site
interaction, for several densities and distinct periodic potential strengths:
(a) $V=0$, (b) $V=-2$, (c) $V=-4$ and (d) $V=-10$.

We then analyze the average single-site entanglement as a function of $U$ in
order to find signatures of the Mott metal-insulator transition within the
superlattice. As shown in Figure 8, our results reveal that entanglement
becomes non-monotonic with the density (crossing between distinct $n$ curves)
whenever the entire chain or a portion of it reaches the Mott-insulator
regime: at $V=0$ (homogeneous finite chain), it happens only for the $n=1$
curve, once this is the only regime for the Mott physics in the absence of the
superlattice potential. In contrast, depending on $V$, the non-monotonicity
with $n$ is observed also for $n<1$, thus detecting the Mott insulator in a
portion (the impurity sites) of the chain, as confirmed by Fig. 7.
Consistently we also see that the interaction for which the curves cross
increases by increasing $V$, supporting the interpretation that the mechanism
behind the Mott insulator phase in superlattices is the competition between
$U$ and $V$. Remarkably DFT properly captures all this physics.

Figure 9: DMRG-DFT deviation for the ground-state energy as a function of the
confinement curvature $k$ for several densities and interactions: (a) $U=0$,
(b) $U=2$, (c) $U=4$ and (d) $U=8$.

In Figure 9 we present the DFT performance in recovering the DMRG ground-state
energies for harmonically confined chains ($V_{i}=k(i-L/2)^{2}$) as a function
of the potential curvature $k$. Although the deviations increase with the
interaction, we find that for typical strongly correlated systems ($U=4$) DFT
energies deviate from DMRG by at most $2.5\%$, what can still be considered as
a fair accuracy.

Figure 10: (a) Density profile for confined lattices, with $L=100$ sites, for
several confinement curvatures $k$. (b) Entanglement as a function of the
confinement curvature $k$: averaging over the entire chain, $\bar{S}$, and
averaging only at the effective chain (central sites with occupation
$n_{i}>0.001$), $\bar{S}_{eff}$. While in both cases DFT reproduces the DMRG
data, one sees that $\bar{S}_{eff}$ is more sensitive to the transitions
between the phases: (i) metallic, (ii) Mott-like insulating, (iii) metallic at
the center of the chain, and (iv) band insulating at the core.

For the density profile we find also a good agreement between the techniques,
as can be seen in Figure 10a. We also observe that the effective chain $-$
defined by the central region where the particles are effectively distributed
($n_{i}>0.001$) $-$ decreases by increasing $k$, while the densities at the
center of the chain increase. This occurs due to the strong potential at the
wings. The density profile also reflects the metallic and insulating phases:
(i) a metallic phase with $n_{i}<1$ at the entire chain, (ii) the Mott-like
insulating phase, where the density at the center is kept fixed at $n_{i}=1$
and a stronger $V$ potential is required to produce $n_{i}>1$, (iii) another
metallic phase with $n_{i}>1$ at the center of the chain; and (iv) a band-
insulator phase, with maximum occupation $n_{i}=2$ at the core.

As shows Figure 10b, the average single-site entanglement is very well
reproduced by DFT calculations in harmonically confined chains. Notice that at
$\bar{S}$ the signatures of the distinct phases are very subtle, therefore in
Ref. [33] the entanglement derivative was used to clearly identify the phases
induced by the harmonical confinement. But numerical derivatives require a
very large number of data to be precise, thus we here propose an alternative
analysis to the transitions, which consists of considering the average single-
site entanglement only at the effective chain,
$\bar{S}_{eff}\equiv\bar{S}(n_{i}>0.001)$, i.e. for the sites that are
effectively occupied. Fig. 10b reveals that $\bar{S}_{eff}$ signs better the
distinct phases: it decreases monotonically with $k$ in the metallic phase
(i); has a local minimum at the Mott-like transition (ii); a local maximum at
the metallic regime (iii); and a monotonic decreasing for the band-insulator
phase (iv).

Table 1: Deviations for the ground-state energy (Eq.(7)), the average entanglement (Eq. (8)), and the density profile (Eq. (9)), for homogeneous chains ($U=8$, $n=0.6$) of distinct sizes $L$, superlattices ($U=8$, $n=0.4$, $V=-2$, $L=36$) with different structures SL, and harmonical confinement ($U=8$, $n=0.5$, $L=100$) for distinct curvatures $k$.  | GS energy | density profile | entanglement
---|---|---|---
| $D\%(e_{0})$ | $\bar{D}\%(n)$ | $D\%(\bar{S})$
Homogeneous |  |  |
$L=10$ | $2.30$ | $3.09$ | $0.43$
$L=50$ | 0.44 | 0.88 | 0.14
$L=100$ | 0.21 | 0.28 | 0.11
Superlattices |  |  |
SL $2:7$ | 1.14 | 12.05 | 2.56
SL $3:6$ | 1.85 | 9.60 | 1.35
SL $4:5$ | 1.51 | 17.19 | 2.92
Harmonical Confinement |  |  |
metal (i), $k=0.002$ | 0.01 | 7.94 | 0.16
Mott-like insulating (ii), $k=0.005$ | 0.12 | 24.40 | 0.02
metal (iii), $k=0.015$ | 0.01 | 3.96 | 0.20
band insulator (iv), $k=0.035$ | 22.17 | 8.84 | 6.62

Finally in Table 1 we present the deviations for a few representative cases of
each of the systems, homogeneous, superlattices and harmonical confinement,
for strongly correlated systems with $U=8$, which represents a challenge to
DFT calculations. In general one finds that the deviations for the density
profile are greater than the deviations for the ground-state energy and
entanglement for all the systems. We attribute this to error cancellations
related to the under and overestimate of DFT for the density profile: while
Eq.(9) will always sum up, thus producing a higher average deviation; Eqs. (7)
and (8) will have smaller and greater contributions, thus on average producing
a small deviation. While for the homogeneous systems there is a clear
hierarchy between the deviations, $D\%(\bar{S})<D\%(e_{0})<\bar{D}\%(n)$, and
all the deviations decrease with $L$; for superlattices and harmonical
confinement the relation among deviations is less trivial and strongly
dependent on the SL structure and the confinement strength considered. For the
superlattices, in general DFT calculations becomes less precise by increasing
the number of impurities in the unit cell ($3:6$ and $4:5$). For the confined
chains, DFT performs better for the metallic phases (i) ($k=0.002$) and (iii)
($k=0.015$). The highest deviations appear for the band-insulator phase (iv)
($k=0.035$). At this (iv) phase the density profile shows essentially only
three values, $n_{i}=0$ at the chain borders, $n_{i}=1$ at the wings of the
potential and $n_{i}=2$ at the core (see Fig. 10a), thus any imprecision on
the DFT calculations has a bigger impact than in a metallic phase whose
density profile is more homogeneously distributed.

##  4 Conclusions

We have performed a comparative analysis between DFT and DMRG calculations,
focusing on their efficacy in describing the electronic properties and quantum
phase transitions of one-dimensional homogenous, superlattices, and
harmonically confined nanostructures.

For homogeneous finite chains, DFT exhibited significant deviations from the
DMRG results for chain sizes $L\lesssim 60$, especially for the ground-state
energy near the half-filling case: $\sim 16\%$ for small chains $L=10$,
although it is less than 1% for $L\sim 100$. This arises from additional
electronic correlations due to Friedel oscillations, which are not very well
described by the local density approximation within DFT. However, far from the
half-filling case, around 20% band filling, the DFT exhibited the best
performance, presenting a maximum deviation of less than 2% for the ground-
state energy for any interaction and chain size. Also, there is a clear
hierarchy between the deviations, $D\%(\bar{S})<D\%(e_{0})<\bar{D}\%(n)$.

For superlattice and confined systems the relation among the deviations is
less trivial $-$ strongly dependent on the superlattice structure and the
confinement strength $-$ and the inaccuracy of the DFT calculation is
amplified, what is justified by the heterogeneous nature of the spatial
distribution of electronic occupancy, representing a further challenge to the
LDA approach. For the superlattices, in general increasing the number of
impurities in the unit cell represents less precision of the DFT calculations.
For the confined chains, DFT performs better for the metallic phases, while
the highest deviations appear for the Mott and band-insulator phases.

Although the DRMG offers an accurate description of the wave function of 1D
systems, it faces computational constraints that impede its scalability to
larger nanostructures: the DRMG calculation is restricted to the growth of the
system entanglement, which depends on the chain length, while the DFT approach
is practicable in arbitrary large systems. Furthermore, DFT results are
generally obtained in shorter computational simulation times, typically on the
order of minutes, while accurate convergence of the DMRG calculation can reach
hours.

This comparative study highlights the importance of leveraging strategies that
combine the capabilities of DFT and DMRG to predict electronic properties
within one-dimensional nanostructures. Future research efforts can focus on
developing hybrid methodologies that synergistically employ the advantages of
these approaches, especially in describing the correlation potential [34],
thus offering a more robust and accurate framework for modeling and
understanding nanoscale systems.

Acknowledgments This research was supported by FAPESP (2021/06744-8;
2023/00510-0; 2023/02293-7; 2021/02342-2), CNPq (403890/2021-7; 140854/2021-5;
306301/2022-9), Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior -
Brasil (CAPES) - Finance Code 001, and by resources supplied by the Center for
Scientific Computing (NCC/GridUNESP) from São Paulo State University.

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## References

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  * Schollwöck [2011] Schollwöck, U.: The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326(1), 96–192 (2011) https://doi.org/10.1016/j.aop.2010.09.012
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