# Finite groups with abelian automorphism groups : A Survey

Rahul Dattatraya Kitture  and  Manoj K. Yadav

###### Abstract.

In this article we present an extensive survey on the developments in the
theory of non-abelian finite groups with abelian automorphism groups, and pose
some problems and further research directions.

###### Key words and phrases:

Miller group, central automorphism, special $p$-group

###### 2010 Mathematics Subject Classification:

20D45, 20D15

##  1\. Introduction

All the groups considered here are finite groups.

Any cyclic group has abelian automorphism group. By the structure theorem for
finite abelian groups, it is easy to see that among abelian groups, only the
cyclic ones have abelian automorphism groups. A natural question, posed by H.
Hilton [22, Appendix, Question 7] in 1908, is:

Can a non-abelian group have abelian group of automorphisms?

An affirmative answer to this question was given by G. A. Miller [36] in 1913.
He constructed a non-abelian group of order $2^{6}$ whose automorphism group
is elementary abelian of order $2^{7}$. Observe that a non-abelian group with
abelian automorphism group must be a nilpotent group of nilpotency class $2$.
Hence it suffices to study the groups of prime power orders. Investigation on
the structure of groups with abelian automorphism groups was initiated by C.
Hopkins [23] in 1927. He, among other things, proved that such a group can not
have a non-trivial abelian direct factor, and if such a group is a $p$-group,
then so is its automorphism group, where $p$ is a prime integer.

Unfortunately, the topic was not investigated for about half a century after
the work of Hopkins. But days came when examples of such odd prime power order
groups were constructed by D. Jonah and M. Konvisser [29] in 1975 and a thesis
was written on the topic by B. E. Earnley [12] during the same year, in which,
attributing to G. A. Miller, a group with abelian automorphism group was named
as ‘Miller group’. Following Earnley, we call a group to be Miller if it is
non-abelian and its automorphism group is abelian. Earnley pointed out that
the former statement of Hopkins is not true as such for $2$-groups. He
constructed Miller $2$-groups admitting a non-trivial abelian direct factor.
However, the statement is correct for odd order groups. He also presented a
generalization of examples of Jonah-Konvisser, which implies that the number
of elements in a minimal generating set for a Miller group can be arbitrarily
large. But he obtained a lower bound on the number of generators, which was
later improved to an optimal bound by M. Morigi [38]. He also gave a lower
bound on the order of Miller $p$-groups, which was again improved to an
optimal bound by M. Morigi [37].

Motivated by the work done on the topic, various examples of Miller groups
were constructed by several mathematicians via different approaches during the
next 20 years, which were mostly special $p$-groups. A $p$-group $G$ is said
to be special if $\operatorname{Z}(G)=G^{\prime}$ is elementary abelian, where
$\operatorname{Z}(G)$ and $G^{\prime}$ denote the center and the commutator
subgroup of $G$ respectively. These include the works by R. Faudree [14], H.
Heineken and M. Liebeck [20], D. Jonah and M. Konvisser [29], B. E. Earnley
[12], H. Heineken [19], A. Hughes [24], R. R. Struik [39], S. P. Glasby [16],
M. J. Curran [8], and M. Morigi [37]. The existence of non-special Miller
groups follows from [29, Remark 2].

The abelian $p$-groups of minimum order, which can occur as automorphism
groups of some $p$-groups were studied by P. V. Hegarty [18] in 1995 and G.
Ban and S. Yu [2] in 1998. More examples of Miller groups were constructed by
A. Jamali [26] and Curran [10].

Neglecting Remark 2 in [29], A. Mahalanobis [33], while studying Miller groups
in the context of MOR cryptosystems, conjectured that Miller $p$-groups are
all special for odd $p$. Again neglecting [29, Remark 2], non-special Miller
$p$-groups were constructed by V. K. Jain and the second author [27], V. K.
Jain, P. K. Rai and the second author [28], A. Caranti [6] and the authors
[31].

The motivation for many of the examples comes from some natural questions or
observations on previously known examples. The construction of examples of
Miller groups of varied nature have greatly contributed to complexify the
structure of a Miller group. The structure of a general Miller group has not
yet been well understood.

We record here the known information about structure of Miller $p$-groups and
their automorphism groups. Let $G$ be a Miller $p$-group, and assume that it
has no abelian direct factor. Then the following hold:

  1. (1)

$|G|\geq p^{6}$ if $p=2$ and $|G|\geq p^{7}$ if $p>2$.

  2. (2)

Minimal number of generators of $G$ is $3$ for $p$ even, and $4$ for $p$ odd.

  3. (3)

The exponent of $G$ is at least $p^{2}$.

  4. (4)

If $|G^{\prime}|>2$, then $G^{\prime}$ has at least two cyclic factors of the
maximum order in its cyclic decomposition.

  5. (5)

If ${\rm Aut}(G)$ is elementary abelian, then $\Phi(G)$ is elementary abelian,
$G^{\prime}\leq\Phi(G)\leq Z(G)$ and at least one equality holds. There are
Miller $p$-groups in which only one equality holds.

  6. (6)

For $p>2$ (resp. $p=2$), the abelian $p$-groups of order $<p^{11}$ (resp.
$<2^{7}$) are not automorphism groups of any $p$-group.

The aim of this article is to present an extensive survey on the developments
in the theory of Miller groups since 1908, and pose some problems and further
research directions. There are, at least, three survey articles on
automorphisms of $p$-groups and related topics (see [21], [34], [35]); the
present one does not overlap with any of them.

We conclude this section with setting some notations for a multiplicatively
written group $G$. We denote by $\Phi(G)$, the Frattini subgroup of $G$. For
an element $x$ of $G$, $\langle x\rangle$ denotes the cyclic subgroup
generated by $x$, and $o(x)$ denotes its order. For subgroups $H,K$ of a group
$G$, $H<K$ (or $K>H$) denotes that $H$ is proper subgroup of $K$. The exponent
of $G$ is denoted by $\exp(G)$. By $\operatorname{C}_{n}$, we denote the
cyclic group of order $n$. For a $p$-group $G$ and integer $i\geq 1$,
$\Omega_{i}(G)$ denotes the subgroup of $G$ generated by those $x$ in $G$ with
$x^{p^{i}}=1$, and $\mho^{i}(G)$ denotes the subgroup generated by $x^{p^{i}}$
for all $x$ in $G$. By ${\operatorname{Inn}}(G)$,
${\operatorname{Autcent}}(G)$ and ${\operatorname{Aut}}(G)$ we denote,
respectively, the group of inner automorphisms, central automorphism, and all
automorphisms of $G$. Let a group $K$ acts on a group $H$ by automorphisms,
then $H\rtimes K$ denotes the semidirect product of $H$ by $K$. All other
notations are standard.

##  2\. Reductions

Starting from the fundamental observations of Hopkins, in this section, we
summarize all the known results (to the best of our knowledge) describing the
structure of Miller groups. Although we preserve the meaning, no efforts are
made to preserve the original statements from the source.

As commented in the introduction, a non-abelian Miller group must be nilpotent
(of class $2$), and therefore it is sufficient to study Miller $p$-groups. The
following result is an easy exercise.

######  Proposition 1 (Hopkins, [23]).

Every automorphism of a Miller group centralizes $G^{\prime}$.

An automorphism of a group $G$ is said to be central if it induces the
identity automorphism on the central quotient $G/\operatorname{Z}(G)$. Note
that ${\operatorname{Autcent}}(G)$, the group of all central automorphisms of
$G$, is the centralizer of ${\operatorname{Inn}}(G)$ in
${\operatorname{Aut}}(G)$.

A group $G$ is said to be purely non-abelian if it has no non-trivial abelian
direct factor. Hopkins made the following important observation:

######  Theorem 2 (Hopkins, [23]).

A Miller $p$-group is purely non-abelian.

Unfortunately, the statement, as such, is not true for $p=2$ as shown by
Earnley [12] (see Theorem 5 below for the correct statement).

In a finite abelian group $G$, there exists a minimal generating set
$\\{x_{1},\ldots,x_{n}\\}$ such that $\langle x_{i}\rangle\cap\langle
x_{j}\rangle=1$ for $i\neq j$. Hopkins observed that Miller groups possess a
set of generators with a property similar to the preceding one. More
precisely, he proved the following result.

######  Theorem 3 (Hopkins, [23]).

If $G$ is a Miller $p$-group, then there exists a set
$\\{x_{1},\ldots,x_{n}\\}$ of generators of $G$ such that

  1. (1)

for $p>2$, $\langle x_{i}\rangle\cap\langle x_{j}\rangle=1$ for all $i\neq j$;

  2. (2)

for $p=2$, $\langle x_{i}\rangle\cap\langle x_{j}\rangle$ is of order at most
$2$, for all $i\neq j$.

In fact the theorem is true for any $p$-group of class $2$, as shown below.
Let $G$ be a $p$-group of class $2$. Choose a set $\\{y_{1},\ldots,y_{k}\\}$
of generators of $G$ with the property

$\prod_{i}o(y_{i})$ is minimum. (*)
Let $p>2$ and $o(y_{i})=p^{n_{i}}$, $1\leq i\leq k$. If $\langle
y_{1}\rangle\cap\langle y_{2}\rangle\neq 1$ then
$y_{2}^{p^{n_{2}-1}}\in\langle y_{1}\rangle$. Assuming that $o(y_{1})\geq
o(y_{2})$, we can write $y_{2}^{p^{n_{2}-1}}=(y_{1}^{p^{n_{2}-1}})^{a}$ for
some integer $a$. Take $y_{2}^{\prime}=y_{2}y_{1}^{-a}$. Then
$\\{y_{1},y_{2}^{\prime},y_{3},\ldots,y_{k}\\}$ is a generating set for $G$
and $o(y_{2}^{\prime})\leq p^{n_{2}-1}<o(y_{2})$, which contradicts (*). The
case $p=2$ can be handled in a similar way.

######  Theorem 4 (Hopkins, [23]).

If $G$ is a Miller $p$-group, then ${\operatorname{Aut}}(G)$ is a $p$-group.

This can be proved easily in the following way. In the case when $G$ is purely
non-abelian, by a result of Adney-Yen [1, Theorem 1],
${\operatorname{Autcent}}(G)$ ($={\operatorname{Aut}}(G)$) has order equal to
$|{\operatorname{Hom}}(G,\operatorname{Z}(G))|$, which is clearly a power of
$p$. Now consider the case when $G$ has an abelian direct factor, which occurs
only when $p=2$ and it must be cyclic of order at least $2^{2}$ (see Theorem
5). Let $G=H\times C_{2^{n}}$, where $H$ is purely non-abelian and $n\geq 2$.
By the main theorem in [4],

$|{\operatorname{Aut}}(G)|=|{\operatorname{Aut}}(H)|\cdot|{\operatorname{Aut}}(C_{2^{n}})|\cdot|{\operatorname{Hom}}(H,C_{2^{n}})|\cdot|{\operatorname{Hom}}(C_{2^{n}},Z(H))|,$

and each factor on the right side has order a power of $2$.

The investigation of structure of Miller groups remained unattained for about
half a century until it was revisited by Earnley [12] in 1974. He pointed out
that a Miller $2$-group can have an abelian direct factor. So the correct form
of Theorem 2 is

######  Theorem 5 (Earnley, [12]).

Let $G$ be a finite $p$-group such that $G=A\times N$ with $A\neq 1$ an
abelian group and $N$ a purely non-abelian group. Then $G$ is a Miller group
if and only if $p=2$ and $A,N$ satisfy the following conditions:

  1. (1)

$A$ is cyclic of order $2^{n}>2$;

  2. (2)

$N$ is a special Miller $2$-group.

######  Theorem 6 (Earnley, [12]).

If $G$ is a Miller $p$-group, then the following hold.

  1. (1)

The exponent of $G$ is greater than $p$;

  2. (2)

If $p>2$, then $Z(G)\cap\Phi(G)$ is non-cyclic.

For the first statement of the preceding theorem, we can assume that $p>2$ and
the proof now follows by noting that if $\exp(G)=p$, then the map $x\mapsto
x^{-1}$ is a non-central automorphism of $G$. The second statement follows
from the following result of Adney-Yen.

######  Theorem 7 (Adney-Yen, [1]).

If $G$ is a $p$-group of class $2$ such that $G^{\prime}$ has only one cyclic
factor of maximum order in the direct product decomposition, then $G$
possesses a non-central automorphism.

Since, in $p$-group of class $2$,
$G^{\prime}\leq\operatorname{Z}(G)\cap\Phi(G)$, the preceding theorem
restricts the structure of the commutator subgroup of a Miller $p$-group.

######  Theorem 8.

If $G$ is a Miller $p$-group, $p$ odd, then $G^{\prime}$ possesses at least
two cyclic factors of maximum order in the direct product decomposition.

This raises a natural question for $p=2$. The analogue of the preceding
theorem for $p=2$ holds except when $|G^{\prime}|=2$. This can be obtained
from the following generalization of Theorem 7.

######  Theorem 9 (Faudree, [13]).

Let $G$ be a $p$-group of class $2$ with the following conditions:

  1. (1)

$G^{\prime}=\langle u\rangle\times{\rm U}$, where
$o(u)=p^{m_{1}}>p^{m^{\prime}}=exp({\rm U}).$

  2. (2)

$[g,h]=u$ and $h^{p^{m_{1}+m^{\prime}}}=1$,

  3. (3)

$m^{\prime\prime}=m^{\prime}$ if $p$ is odd, and
$m^{\prime\prime}=\mbox{max}(1,m^{\prime})$ if $p=2$.

Let $H=\langle g,h\rangle$ and $L=\\{x\in G:[g,x],[h,x]\in{\rm U}\\}$. Then
$G=HL$ and the map

$g\mapsto gh^{p^{m^{\prime\prime}}},\hskip 14.22636pth\mapsto h,\hskip
14.22636ptx\mapsto x\;\;(x\in L)$

defines an automorphism of $G$ which centralizes $Z(G)$.

Notice that if $p=2$ and the exponent of $G^{\prime}$ is at least $4$, then
the automorphism defined in the preceding theorem is non-central, and
therefore $G$ is not Miller. But if $|G^{\prime}|=2$, then the theorem is no-
longer applicable to produce a non-central automorphism of $G$. So the
following question remains open.

Question 1. Can a finite $2$-group $G$ with $G^{\prime}$ cyclic of order $2$
be Miller?

Earnley obtained lower bounds for the order and the minimum number of
generator of a Miller group, which were later sharpened to the optimal level
by Morigi [37, 38]

######  Theorem 10.

Let $G$ be a Miller $p$-group.

  1. (1)

For any prime $p$, $G$ is generated by at least $3$ elements. (Earnley, [12])

  2. (2)

If $p$ is odd, then $G$ is generated by at least $4$ elements. (Morigi,[37])

The example of a Miller $2$-group constructed by Miller is minimally generated
by $3$ elements (see Section 3 (3.1)). For $p$ odd, Jonah-Konvisser
constructed Miller $p$-groups which are minimally generated by $4$ elements
(see Section 4 (4.3) for more details).

######  Theorem 11.

Let $G$ be a Miller $p$-group.

  1. (1)

For any prime $p$, $|G|\geq p^{6}$. (Earnley, [12])

  2. (2)

If $p$ is odd, then $|G|\geq p^{7}$ and there exists a Miller group of order
$p^{7}$. (Morigi, [38])

Again the Miller $2$-group constructed by Miller is of order $2^{6}$ having
automorphism group of order $2^{7}$. Morigi constructed a special Miller
$p$-group $G$ of order $p^{7}$ with $\operatorname{Aut}(G)$ elementary abelian
of order $p^{12}$. In fact, it is one among an infinite family of Miller
groups constructed by Morigi (see Section 4 (4.6) for more details).

The question whether an abelian $p$-group of order smaller than $p^{12}$ for
an odd prime $p$ can occur as the automorphism group of a $p$-group, was
addressed by Hegarty [18] and Ban and Yu [2] independently. They proved

######  Theorem 12.

For $p$ odd, there is no abelian $p$-group of order smaller than $p^{12}$
which can occur as the automorphism group of a $p$-group.

Finally we state some results on the structure of ${\operatorname{Aut}}(G)$
for a Miller group $G$. Many known examples of Miller groups are special
$p$-groups. Certainly for such groups, the automorphism group is elementary
abelian. The following problem appears as an old problem in [3, problem 722].

Problem 2. Study the $p$-groups with elementary abelian automorphism groups.

Let $G$ be a $p$-group. If ${\operatorname{Aut}}(G)$ is elementary abelian,
then so is $G/Z(G)$; hence $G^{\prime}\leq\Phi(G)\leq Z(G)$. In fact, it is
interesting to see that at least one equality always holds.

######  Theorem 13 (Jain-Rai-Yadav, [28]).

Let $G$ be a $p$-group, $p$ odd, such that ${\operatorname{Aut}}(G)$ is
elementary abelian. Then $\Phi(G)$ is elementary abelian, and one of the
following holds:

  1. (1)

$Z(G)=\Phi(G)$.

  2. (2)

$G^{\prime}=\Phi(G)$.

Jain-Rai-Yadav [28] constructed $p$-groups with elementary abelian
automorphism group, in which exactly one of the above two conditions holds.

For $2$-groups with elementary abelian automorphism groups, there are
analogous necessary conditions, stated below. Since a Miller $2$-group can
have abelian (cyclic) direct factor, we consider two cases.

An abelian $p$-group of type $(p^{n},p,\ldots,p)$ with $n>1$ is called a ce-
group. If $G=A\times B$ with $A\cong\operatorname{C}_{p^{n}}$ ($n>1$) and
$B\cong\operatorname{C}_{p}\times\cdots\times\operatorname{C}_{p}$, then call
$A$ cyclic part and $B$ elementary part of $G$.

######  Theorem 14 (Jafari, [25]).

Let $G$ be a purely non-abelian $2$-group. Then ${\operatorname{Autcent}}(G)$
is elementary abelian if and only if one of the following holds.

  1. (1)

$G/G^{\prime}$ is of exponent $2$.

  2. (2)

$Z(G)$ is of exponent $2$.

  3. (3)

${\operatorname{gcd}}\big{(}\exp(G/G^{\prime}),\exp(Z(G))\big{)}=4$ and
$G/G^{\prime},Z(G)$ are ce-groups such that elementary part of $Z(G)$ is
contained in $G^{\prime}$ and there is an element $z$ of order $4$ in cyclic
part of $Z(G)$ with $zG^{\prime}$ lying in cyclic part of $G/G^{\prime}$
satisfying $o(zG^{\prime})=\exp(G/G^{\prime})/2$.

In particular, if $G$ is a purely non-abelian $2$-group with
${\operatorname{Aut}}(G)$ elementary abelian, then one of the above three
conditions holds. The example constructed by Miller shows that it satisfies
only condition (3). Jain-Rai-Yadav [28] constructed purely non-abelian Miller
$2$-groups which satisfy only condition (1) or only condition (2). Finally,
for $2$-groups $G$ with abelian direct factor, the following theorem gives
necessary conditions for ${\rm Aut}(G)$ to be elementary abelian.

######  Theorem 15 (Karimi-Farimani, [30]).

Let $G=A\times N$ with $A$ cyclic $2$-group and $N$ purely non-abelian
$2$-group of class $2$. Then ${\rm Autcent}(G)$ is elementary abelian if and
only if

  1. (1)

$|A|$ is $4$ or $8$.

  2. (2)

$N$ is special $2$-group.

With the notations of the preceding theorem, if $G$ is Miller then (1) and (2)
hold.

Theorem 5 tells us that a Miller $p$-group, $p$ odd, can not admit a non-
abelian direct factor. It is natural to ask whether a Miller $p$-group can
occur as a direct product of non-abelian groups. This situation has been
considered by Curran [10]. He determined necessary and sufficient conditions
on a direct product $H\times K$ to have abelian automorphism group. Note that
for any non-trivial group $H$, $\operatorname{Aut}(H\times H)$ is non-abelian.
Before we state the result of Curran, we set some notations.

Let $H$ be a $p$-group of class $2$.

  1. (1)

Let $a,b,c,d$ denote the exponents of $H/H^{\prime}$, $H/\operatorname{Z}(H)$,
$\operatorname{Z}(H)$ and $H^{\prime}$ respectively.

  2. (2)

If $H^{\prime}$ and $\operatorname{Z}(H)$ have the same rank, then define
$d_{s}$ to be the largest integer ($\leq\exp(\operatorname{Z}(H)$) such that
$\Omega_{d_{s}}(H^{\prime})=\Omega_{d_{s}}(Z(H))$.

  3. (3)

If $H/\operatorname{Z}(H)$ and $H/H^{\prime}$ have the same rank, then define
$b_{t}$ to be the largest integer ($\leq\exp(H/H^{\prime})$) such that
$\Omega_{b_{t}}(H/\operatorname{Z}(H))\cong\Omega_{b_{t}}(H/H^{\prime})$.

  4. (4)

Let $\Gamma^{i}(H)$ denote the subgroup of $H$ with
$\Gamma^{i}(H)/H^{\prime}=\mho^{i}(H/H^{\prime})$.

Replacing $H$ by $K$ in (1)-(4), the corresponding terms
$a^{\prime},b^{\prime},c^{\prime},d^{\prime}$, $d^{\prime}_{s^{\prime}}$,
$b^{\prime}_{t^{\prime}}$ and $\Gamma^{i}(K)$ are similarly defined. For
simplicity, we denote by $r(A)$, the rank of an abelian group $A$. With this
setting, we have

######  Theorem 16 (Curran, [10]).

Let $G=H\times K$, where $H,K$ are $p$-groups of class $2$ with no common
direct factor. Then ${\operatorname{Aut}}(G)$ is abelian if and only if
${\operatorname{Aut}}(H)$ and ${\operatorname{Aut}}(K)$ are abelian and one of
the following holds:

  1. (1)

$\operatorname{Z}(H)=H^{\prime}$ and $\operatorname{Z}(K)=K^{\prime}$.

  2. (2)

$\operatorname{Z}(H)>H^{\prime}$ and $\operatorname{Z}(K)=K^{\prime}$, where
$r(\operatorname{Z}(H))=r(H^{\prime})$, $a^{\prime}\leq d_{s}\leq a\leq c$ and
$\Omega_{a}(Z(H))\leq\Gamma^{c^{\prime}}(H)$.

  3. (3)

$\operatorname{Z}(H)=H^{\prime}$ and $\operatorname{Z}(K)>K^{\prime}$, where
$r(\operatorname{Z}(K))=r(K^{\prime})$, $a\leq d^{\prime}_{s^{\prime}}\leq
a^{\prime}\leq c^{\prime}$ and
$\Omega_{a^{\prime}}(\operatorname{Z}(K))\leq\Gamma^{c}(K)$.

  4. (4)

$\operatorname{Z}(H)>H^{\prime}$ and $\operatorname{Z}(K)>K^{\prime}$, where
$r(\operatorname{Z}(H))=r(H^{\prime})$, $r(\operatorname{Z}(K))=r(K^{\prime})$
and $a=d_{s}=d^{\prime}_{s^{\prime}}=a^{\prime}$.

  5. (5)

$\operatorname{Z}(H)>H^{\prime}$ and $\operatorname{Z}(K)=K^{\prime}$, where
$r(H/\operatorname{Z}(H))=r(H/H^{\prime})$, $a^{\prime}\leq b_{t}\leq c\leq a$
and $\Omega_{a^{\prime}}(\operatorname{Z}(H))\leq\Gamma^{c}(H)$.

  6. (6)

$\operatorname{Z}(H)=H^{\prime}$ and $\operatorname{Z}(K)>K^{\prime}$, where
$r(K/\operatorname{Z}(K))=r(K/K^{\prime})$, $a\leq b^{\prime}_{t^{\prime}}\leq
c^{\prime}\leq a^{\prime}$ and
$\Omega_{a}(\operatorname{Z}(K))\leq\Gamma^{c^{\prime}}(K)$.

  7. (7)

$\operatorname{Z}(H)>H^{\prime}$ and $\operatorname{Z}(K)>K^{\prime}$, where
$r(H/\operatorname{Z}(H))=r(H/H^{\prime})$,
$r(K/\operatorname{Z}(K))=r(K/K^{\prime})$ and
$c=b_{t}=b^{\prime}_{t^{\prime}}=c^{\prime}$.

######  Remark 17.

Observe that in the above theorem, if ${\operatorname{Aut}}(H\times K)$ is
abelian then either $H^{\prime}$ and $\operatorname{Z}(H)$ have the same rank
or $H/H^{\prime}$ and $H/Z(H)$ have the same rank; the same is true for the
other component $K$.

An analogous problem for central product may be stated as follows.

Problem 3. Find necessary and/or sufficient condition such that central
product of two Miller groups is again a Miller group.

##  3\. Examples of Miller $2$-groups

In this section, we discuss examples of Miller $2$-groups in chronological
order.

(3.1) As mentioned above several times, the first example of a Miller
$2$-group was constructed by Miller himself, which comes as a semi-direct
product of the cyclic group of order $8$ by the dihedral group of order $8$,
and presented by

$\displaystyle G_{1}$ $\displaystyle=C_{8}\rtimes D_{8}=\langle x,y,z\mid
x^{8},y^{4},z^{2},zyz^{-1}=y^{-1},yxy^{-1}=x^{5},zxz^{-1}=x\rangle.$

Miller proved that each coset of $Z(G_{1})$ is invariant under every
automorphism of $G_{1}$ and that every automorphism has order dividing $2$.
Thus ${\operatorname{Aut}}(G_{1})$ is elementary abelian, and it can be shown
that its order is $2^{7}$.

By Theorem 11, the order of a Miller $2$-group is at least $64$. Having an
example of order $64$, a natural idea which peeps in ones mind is to explore
groups of order $64$ to find more Miller groups. This was done by Earnley
[12], who proved that there are (exactly) two more Miller groups of the
minimal order, which are presented as follows.

$\displaystyle G_{2}$ $\displaystyle=(C_{4}\rtimes C_{4})\rtimes C_{4}=\langle
x,y,z\mid yxy^{-1}=x^{-1},zxz^{-1}=xy^{2},zyz^{-1}=y\rangle,$ $\displaystyle
G_{3}$ $\displaystyle=(C_{4}\times C_{4}\times C_{2})\rtimes C_{2}=\langle
x,y,z,t\mid x^{4},y^{4},z^{2},t^{2},xy=yx,xz=zx,yz=zy,$ $\displaystyle\hskip
170.71652pttxt^{-1}=xy^{2},tyt^{-1}=tz,t^{-2}zt=z\rangle.$

Note that in all the groups $G_{1},G_{2}$ and $G_{3}$, the center and Frattini
subgroups coincide and are elementary abelian of order $8$, whereas the
commutator subgroup is elementary abelian of order $4$. Further,
${\operatorname{Aut}}(G_{2})$ and ${\operatorname{Aut}}(G_{3})$ are elementary
abelian of order $2^{9}$.

A generalization of $G_{1}$ appears, as an exercise, in the book [32, Exercise
46, Page 237] by Macdonald, written in 1970. For $n\geq 3$, the group is
presented as follows.

$\displaystyle G_{1,n}=\langle a,b,c\mid$ $\displaystyle
a^{2^{n}}=b^{4}=c^{2}=1,b^{-1}ab=a^{1+2^{n-1}},\hskip
5.69054ptc^{-1}bc=b^{-1},\hskip 5.69054pt[c,a]=1\rangle.$

The group $G_{1,n}$ is of order $2^{n+3}$ and its automorphism group is an
abelian $2$-group of type $(2^{n-2},2,2,2,2,2,2)$. In 1982, Struik [39],
independently, obtained the same example with a different presentation.

A generalization of $G_{2}$ and $G_{3}$ has also been obtained by Glasby [16],
which is described as follows. For $n\geq 3$, let $G_{2,n}$ denotes the
$2$-group of class $2$ with generators $x_{1},x_{2},\cdots,x_{n}$ with
following additional relations:

$\displaystyle x_{i}^{4}=1,\,\,(1\leq i\leq
n),\,\,[x_{i},x_{n}]=x_{i+1}^{2},\,\,(1\leq i\leq n-1),$

and set $[x_{k},x_{l}]=1$ in the remaining cases. Here ${\rm Aut}(G_{2,n})$ is
elementary abelian group of order $2^{2n}$. If $n=3$ then $G_{2,n}$ is
isomorphic to $G_{2}$.

Again for $n\geq 3$, let $G_{3,n}$ denotes the $2$-group of class $2$ with
generators $y_{0}$, $y_{1}$, $\ldots$, $y_{n}$ with following additional
relations:

$\displaystyle y_{0}^{2}=y_{i}^{4}=y_{n}^{2}=1,\,\,(1\leq i\leq
n-1),\,\,[y_{i},y_{n}]=y_{i+1}^{2},\,\,(1\leq i\leq n-2)$

and set $[y_{k},y_{l}]=1$ in the remaining cases. Here also ${\rm
Aut}(G_{3,n})$ is elementary abelian group of order $2^{2n}$. If $n=3$ then
$G_{3,n}$ is isomorphic to $G_{3}$.

(3.2) It should be noted that, in 1974, Jonah and Konvisser constructed
special Miller groups of order $p^{8}$, which were generalized to an infinite
family of Miller $p$-groups by Earnley, and those groups include the case
$p=2$ too (see Section 4 (4.3) and (4.4)).

(3.3) Heineken and Liebeck (see Section 4 (4.2) for details) proved that given
a finite group $K$, there exists a special $p$-group $G$, $p$ odd, with ${\rm
Aut}(G)/{\rm Autcent}(G)\cong K.$ In particular, if $K=1$, then the
corresponding group $G$ is Miller $p$-group.

In 1980, A. Hughes [24] proved that one can construct a special $2$-group as
well with the above property. The method of Hughes is a modification of the
graph theoretic method of Heineken-Liebeck. It should be noted that the method
of Heineken-Liebeck uses digraphs, whereas that of Hughes considers graphs,
and is described below.

Let $K$ be any finite group and associate to $K$ a connected graph $D(K)$
which satisfies following conditions:

  1. (1)

Each vertex of the graph has degree at least $2$.

  2. (2)

Every cycle in the graph contains at least $4$ vertices.

  3. (3)

${\rm Aut}(D(K))\cong K$.

Associate a special $2$-group $G$ to $D(K)$ as follows. If the graph has $n$
vertices $v_{1},\ldots,v_{n}$, then consider the free group $F_{n}$ on
$x_{1},\ldots,x_{n}$. Let $R$ be the normal subgroup of $F_{n}$ generated by
$x_{i}^{2}$, $[x_{i},[x_{j},x_{k}]]$ (for all $i,j,k$) and $[x_{r},x_{s}]$
whenever the vertices $v_{r}$ and $v_{s}$ are adjacent. Define $G$ to be the
group $F_{n}/R$; it is a special $2$-group of order $2^{n+\binom{n}{2}-e}$,
where $n$ is the number of generators of $G$ (so $|G/G^{\prime}|=2^{n}$) and
$e$ is the number of edges of the graph. It turns out that ${\rm Aut}(G)/{\rm
Autcent}(G)\cong K$ (see [24] for proof).

(3.4) In 1987, Curran [8] studied automorphisms of semi-direct product, and
suggested a method to construct many more examples of Miller $2$-groups
similar to $G_{1}$. We describe the method briefly and see that the above
$G(n)$ can be constructed by this method. Let $A=\langle a\rangle$ be a cyclic
group of order $2^{n}$, $n\geq 3$ and $N$ a special $2$-group acting on $A$ in
the following way: a maximal subgroup $J\leq N$ acts trivially, and any $x\in
N\setminus J$ acts by

$xax^{-1}=a^{1+2^{n-1}}.$

Let $G=A\rtimes N$ be the semi-direct product of $A$ and $N$ with this action.
Then we get

######  Theorem 18 (Curran, [8]).

Let $G=A\rtimes N$ be as above along with the conditions:

  1. (1)

$A\times J$ is characteristic in $A\rtimes N$.

  2. (2)

Any automorphism of $N$ leaving $J$ invariant is central automorphism of $N$.

Then $A\rtimes N$ is a Miller group.

To elaborate, consider $A=\langle a\rangle$, the cyclic group of order
$2^{n}$, $n\geq 3$ and $N=\langle b,c\mid b^{4},c^{2},[b,c]=b^{2}\rangle$, the
dihedral group of order $8$. Consider the action of $N$ on $A$ by

$b^{-1}ab=a^{1+2^{n-1}},\hskip 14.22636ptc^{-1}ac=a.$

Define $G=A\rtimes N$, the semi-direct product with this action. Note that
$Z(G)=\langle a^{2},b^{2}\rangle=\Phi(G)$, and $G/\Phi(G)$ is elementary
abelian of order $8$.

Take $J=\langle b^{2},c\rangle$, the largest subgroup of $N$ acting trivially
on $A$. Then $A\times J$ is characteristic in $A\rtimes N$, since it is the
unique abelian subgroup of index $2$ (if there were more than one abelian
subgroups of index $2$, then center would have index $4$). Also, in $N$, the
subgroup $\langle b\rangle$ is characteristic. Hence if
$\varphi\in{\operatorname{Aut}}(N)$ leaves $J$ invariant, then it leaves
invariant the subgroup $J\cap\langle b\rangle=\langle b^{2}\rangle$ (the
center of $N$), and one can see that $\varphi$ is central automorphism of $N$.
The group $A\rtimes N$ is therefore a Miller group by Theorem 18. Note that
this group is isomorphic to $G(n)$ described above.

(3.5) After a considerable time gap, Jamali [26] in 2002 constructed the
following infinite family of Miller $2$-groups in which, the number of
generators and the exponent of the group can be arbitrarily large. For
integers $m\geq 2$ and $n\geq 3$, let $G_{n}(m)$ be the group generated by
$a_{1},\ldots,a_{n},b$, subject to following relations:

$\displaystyle a_{1}^{2}=a_{2}^{2^{m}}=a_{i}^{4}=1\hskip 14.22636pt(3\leq
i\leq n)$ $\displaystyle a_{n-1}^{2}=b^{2},$
$\displaystyle[a_{1},b]=[a_{i},a_{j}]=1\;\;\;(1\leq i<j\leq n),$
$\displaystyle[a_{n},b]=a_{1},[a_{i-1},b]=a_{i}^{2}\;\;\;(3\leq i\leq n).$

The group $G_{n}(m)$ has order $2^{2n+m-2}$ and its automorphism group is
abelian of type $(\underbrace{2,2,\ldots,2}_{n^{2}},2^{m-2})$. For $n=3$ and
$m=2$, the group $G_{3}(2)$ is isomorphic to $G_{3}$ (see Section 3(3.1)).

##  4\. Examples of Miller $p$-groups, $p$-odd

This section, a lifeline for Miller groups in a sense, presents evolution of
the topic. We’ll see the influence of examples of varied nature on Miller
groups towards understanding the structure. Some examples of Miller groups
occurred in other related contexts without any pointer to the topic. The
deriving force behind many of the examples comes from natural questions on
previously known Miller groups or certain natural optimistic expectations on
the structure of such groups. The evolution process certainly helped, although
minimally, in studying the structure of Miller groups, especially, in turning
down the natural optimistic expectations.

During 1971 to 1979, there were three occasions, in which certain $p$-groups
were constructed with a specific property, and these groups turned out to be
Miller groups. The motive of the construction of these groups had no obvious
connection with Miller groups.

(4.1) We now start the discussion of the (possibly) first example of Miller
group of odd order. It was conjectured that a finite group in which every
element commutes with its epimorphic image is abelian. It was disproved by R.
Faudree [14] in 1971. He constructed a non-abelian group $G$ in which every
element commutes with its epimorphic image. The group $G$ is a special
$p$-group, and therefore one can easily deduce that ${\rm Aut}(G)$ is abelian.
The group $G$ is described as follows.

Let $G=\langle a_{1},a_{2},a_{3},a_{4}\rangle$ be the $p$-group of class $2$
with following additional relations:

$\displaystyle[a_{1},a_{2}]=a_{1}^{p},\,[a_{1},a_{3}]=a_{3}^{p},\,[a_{1},a_{4}]=a_{4}^{p},$
$\displaystyle[a_{2},a_{3}]=a_{2}^{p},\,[a_{2},a_{4}]=1,\,[a_{3},a_{4}]=a_{3}^{p}.$

The group $G$ is a special $p$-group of order $p^{8}$ and $Z(G)=G^{\prime}$ is
elementary abelian of order $p^{4}$. If $p$ is odd, then ${\rm Aut}(G)$ is
elementary abelian of order $p^{16}$.

(4.2) In 1974, Heineken-Liebeck [20], constructed a $p$-group $G$ of class
$2$, $p$ odd, for a given finite group $K$ such that
${\operatorname{Aut}}(G)/{\operatorname{Autcent}}(G)\cong K$. Associate to the
finite group $K$, a connected digraph (directed graph) $X$ as follows. If $K$
is cyclic of order $>2$, then take $X$ to be the cyclic digraph with $|K|$
vertices. In the remaining cases, we associate a digraph $X$ with following
conditions:

(1) $X$ is strongly connected,

(2) Any two non-simple vertices are not adjacent,

(3) Every vertex belongs to a (directed) cycle of length at least $5$,

(4) Every non-simple vertex has at least two outgoing edge,

(5) ${\rm Aut}(X)\cong K$,
where by a simple vertex we mean a vertex with exactly one incoming and
exactly one outgoing edge. Associate to such $X$, a $p$-group $G_{X}$ of class
$2$ as follows. If $X$ has $n$ vertices, we take $n$ generators for $G_{X}$.
If the vertex $i$ has outgoing edges to $j_{1},\ldots,j_{r}$ precisely, then
we put the relation $x_{i}^{p}=[x_{i},x_{j_{1}}\cdots x_{j_{r}}]$. Finally we
put $[[x_{i},x_{j}],x_{k}]=1$ for all $i,j,k$, making $G_{X}$ of class $2$. It
is easy to show that $G_{X}$ is a special $p$-group of order
$p^{n+\binom{n}{2}}$.

The conditions (1)-(4) imply that ${\rm Aut}(G)/{\rm Autcent}(G)$ is
isomorphic to the automorphism group of $X$, which is isomorphic to $K$ (by
(5)).

Before proceeding for the example of Miller $p$-group by this method, we make
some comments. If $|K|\geq 5$, then there is a systematic procedure to
construct graph $X$ satisfying conditions (1) to (5); it is obtained by a
specific subdivision of a Cayley digraph of $K$. If $|K|<5$, then one
constructs $X$ satisfying (1)-(5) by some ad-hoc method.

For $K=1$, the following digraph satisfies conditions (1)-(5).

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$3$$4$$5$$6$$7$$8$$9$

The group $G_{X}$ associated to this digraph is a special Miller $p$-group of
order $p^{9+\binom{9}{2}}=p^{45}$, $p$ odd.

(4.3) The existence of Miller $2$-groups of order $2^{6}$ served as a base to
the belief that $p^{45}$ is too big to be a minimal order for a Miller
$p$-group for odd $p$. This motivated Jonah and Konvisser [29] to construct
Miller $p$-groups of smaller orders. In 1975, they constructed, for each prime
$p$, $p+1$ non-isomorphic Miller $p$-groups of order $p^{8}$ as described
below. Let $\lambda=(\lambda_{1},\lambda_{2})$ be a non-zero vector with
entries in field $\mathbb{F}_{p}$ and $G_{\lambda}=\langle
a_{1},a_{2},b_{1},b_{2}\rangle$ be the $p$-group of class $2$ with following
additional relations:

$\displaystyle
a_{1}^{p}=[a_{1},b_{1}],\;a_{2}^{p}=[a_{1},b_{1}^{\lambda_{1}}b_{2}^{\lambda_{2}}],\;b_{1}^{p}=[a_{2},b_{1}b_{2}],$
$\displaystyle b_{2}^{p}=[a_{2},b_{2}],\;[a_{1},a_{2}]=[b_{1},b_{2}]=1.$

The proof of the fact that $G_{\lambda}$ is Miller group is very elegant. We
briefly describe the idea here as it will be useful in later discussions.

Fix a non-zero vector $\lambda$ and write $G=G_{\lambda}$. Note that $G$ is
special $p$-group of order $p^{8}$ and $G^{\prime}$ is of rank $4$ with
generators $[a_{i},b_{j}]$, $i,j=1,2$.

The subgroups $A=\langle a_{1},a_{2},G^{\prime}\rangle$ and $B=\langle
b_{1},b_{2},G^{\prime}\rangle$ are the only abelian normal subgroups such that
$[A:G^{\prime}]=p^{2}=[B:G^{\prime}]$; hence they are permuted by every
automorphism of $G$. There exists $x\in A$ such that $A^{p}\leq[x,G]$, but
there is no $y\in B$ with $B^{p}\leq[y,G]$; hence $A,B$ are characteristic in
$G$. The only $x\in A$ with $A^{p}\leq[x,G]$ are $a_{1}^{i}z$ with $p\nmid i$
and $z\in G^{\prime}$; they generate characteristic subgroup $\langle
a_{1},G^{\prime}\rangle$. Similarly, those $x\in A$ with $B^{p}\leq[x,G]$
generate a characteristic subgroup, namely $\langle a_{2},G^{\prime}\rangle$
and those $x\in B$ with $x^{p}\in[x,G]$ generate the characteristic subgroup
$\langle b_{2},G^{\prime}\rangle$. Thus, if $\varphi\in{\rm Aut}(G)$, then

$\varphi(a_{1})\equiv a_{1}^{i},\hskip 14.22636pt\varphi(a_{2})\equiv
a_{2}^{j},\hskip 14.22636pt\varphi(b_{1})\equiv
b_{1}^{k_{1}}b_{2}^{k_{2}},\hskip 14.22636pt\varphi(b_{2})\equiv
b_{2}^{l}\pmod{G^{\prime}},$

where $i,j,l$ are integers not divisible by $p$, and $(k_{1},k_{2})\neq(0,0)$.
Using relations in $G$, it is now easy to deduce that $i=j=k_{1}=l=1$ mod $p$
and $k_{2}=0$. Therefore every automorphism of $G$ is central. Since
$G^{\prime}=Z(G)$, it follows that ${\rm Aut}(G)$ is abelian.

The groups $G_{\lambda}$ and $G_{\mu}$, for non-zero vectors
$\lambda,\mu\in\mathbb{F}_{p}\times\mathbb{F}_{p}$, are isomorphic only if
these vectors are dependent.

######  Remark 19.

Note that in the group $G_{\lambda}$, replacing, in the power relations, all
the $p$-th power by $p^{k}$-th power, it can be shown by the same arguments
that the resulting groups are still Miller. If $k>1$, then
$G_{\lambda}^{\prime}=Z(G_{\lambda})$ and has exponent $p^{k}>p$; hence
$G_{\lambda}$ is a non-special Miller group.

We must emphasize the negligence of the preceding remark [29, Remark 2] by the
authors of many recent papers claiming the existence of non-special Miller
groups was not known in the literature until recently.

(4.4) Examples of Jonah and Konvisser were generalized by Earnley [12], during
the same year, by increasing number of generators, in the following way. Fix
$n\geq 2$ and a non-zero $n$-tuple $(\lambda_{1},\ldots,\lambda_{n})$ with
$\lambda_{i}\in\mathbb{F}_{p}$. Let $G_{\lambda}=\langle
x_{1},x_{2},y_{1},y_{2},\ldots,y_{n}\rangle$ be the $p$-group of class $2$
with the following additional relations:

$\displaystyle x_{1}^{p}=[x_{1},y_{1}],\hskip
14.22636ptx_{2}^{p}=[x_{1},y_{1}^{\lambda_{1}}\ldots y_{n}^{\lambda_{n}}],$
$\displaystyle y_{i}^{p}=[x_{2},y_{i}y_{i+1}],\hskip
8.53581pti=1,2,\ldots,n-1,$ $\displaystyle
y_{n}^{p}=[x_{2},y_{n}],[x_{1},x_{2}]=[y_{i},y_{j}]=1\hskip 8.53581pt\mbox{
for all }i,j.$

It can be shown easily that $G_{\lambda}^{\prime}=Z(G_{\lambda})$ is
elementary abelian and generated by the $2n$ elements $[x_{i},y_{j}]$ for
$i=1,2$ and $j=1,2,\ldots,n$; hence order of $G_{\lambda}$ is
$p^{(n+2)+2n}=p^{2+3n}$.

For ${\operatorname{Aut}}(G_{\lambda})$ to be abelian, the obvious necessary
condition is that every automorphism should be central, but since
$G_{\lambda}$ is special, this is sufficient too. We briefly describe the
beautiful linear algebra techniques evolved by Earnley to prove that every
automorphism of $G_{\lambda}$ is central as, with a little variation, these
techniques have been used in different contexts by Morigi [37], Hegarty [18]
and Earnley himself.

For simplicity, fix non-zero vector $\lambda$ and write $G=G_{\lambda}$. Let
$f:G/G^{\prime}\rightarrow G^{\prime}$ denote the map $xG^{\prime}\mapsto
x^{p}$, which is a homomorphism for odd $p$. An automorphism $\alpha$ of
$G/G^{\prime}$ determines its action on $G^{\prime}$ by

$\hat{\alpha}:G^{\prime}\rightarrow G^{\prime},\hskip
14.22636pt\hat{\alpha}([x,y])=[\alpha(xG^{\prime}),\alpha(yG^{\prime})].$

The automorphism $\alpha$ of $G/G^{\prime}$ is induced by an automorphism
$\varphi$ of $G$ if and only if $\hat{\alpha}\circ f=f\circ\alpha$, i.e. the
following diagram commutes:

$G/G^{\prime}$$G/G^{\prime}$$G^{\prime}$.$G^{\prime}$$f$$f$$\alpha$$\hat{\alpha}$

Consequently, every automorphism of $G$ is central if and only if identity is
the only automorphism of $G/G^{\prime}$ which fits in the above commutative
diagram.

Note that $G/G^{\prime}$ and $G^{\prime}$ can be considered as vector spaces
over $\mathbb{F}_{p}$; hence the automorphisms $\hat{\alpha}$ and $\alpha$ in
the above diagram can be considered as (invertible) linear maps. However, if
$p=2$, the map $f$ may not be linear. But, in the group $G$ under
consideration, consider the abelian subgroups $A=\langle
x_{1},x_{2},G^{\prime}\rangle$ and $B=\langle
y_{1},\ldots,y_{n},G^{\prime}\rangle$; they generate $G$ and the restriction
of $f$ to $A/G^{\prime}$ and $B/G^{\prime}$ are homomorphisms, and therefore
linear.

Next, $A$ and $B$ are the only abelian subgroups containing $G^{\prime}$ such
that modulo $G^{\prime}$ each of them have order at least $p^{2}$. For $n>2$,
both $A$ and $B$ are characteristic as their orders are different. But even
for $n=2$, they are characteristic, since, in this case, $A$ contains an
element $t$ such that $A^{p}\leq[t,G]$, but $B$ has no element with similar
property.

Thus, consider an automorphism $\alpha=(\alpha_{1},\alpha_{2})$ of
$G/G^{\prime}=A/G^{\prime}\oplus B/G^{\prime}$, where
$\alpha_{1}\in{\operatorname{Aut}}(A/G^{\prime})$ and
$\alpha_{2}\in{\operatorname{Aut}}(B/G^{\prime})$. As a vector space,
$G^{\prime}=[A,B]$ is isomorphic to $A/G^{\prime}\otimes B/G^{\prime}$, the
tensor product of $A/G^{\prime}$ and $B/G^{\prime}$. Hence the automorphism
induced by $\alpha$ on $G^{\prime}$ is nothing but
$\alpha_{1}\otimes\alpha_{2}$. Then $\alpha$ is induced by an automorphism of
$G$ if and only if the following diagrams commute:

$A/G^{\prime}$$A/G^{\prime}$$G^{\prime}$$G^{\prime}$$f$$f$$\alpha_{1}$$\alpha_{1}\otimes\alpha_{2}$$B/G^{\prime}$$B/G^{\prime}$$G^{\prime}$$G^{\prime}$$f$$f$$\alpha_{2}$$\alpha_{1}\otimes\alpha_{2}$

For the first diagram, consider the following ordered bases:

$\displaystyle\\{x_{1}G^{\prime},x_{2}G^{\prime}\\}\mbox{ for
}A/G^{\prime}\hskip 8.53581pt\mbox{ and }\hskip
8.53581pt\\{[x_{1},y_{1}],[x_{2},y_{1}],\ldots,[x_{1},y_{n}],[x_{2},y_{n}]\\}\mbox{
for }G^{\prime}.$

Also for the second diagram, consider the following bases:

$\displaystyle\\{y_{1}G^{\prime},\ldots,y_{n}G^{\prime}\\}\mbox{ for
}B/G^{\prime}\hskip 5.69054pt\mbox{ and }\hskip
5.69054pt\\{[x_{1},y_{1}],\ldots,[x_{1},y_{n}],[x_{2},y_{1}],\ldots,[x_{2},y_{n}]\\}\mbox{
for }G^{\prime}.$

The matrix of $f$ with respect to these bases can be easily written from the
power-commutator relations in $G$. Writing the matrices of
$\alpha_{1},\alpha_{2},\alpha_{1}\otimes\alpha_{2}$ with respect to these
bases, one expresses the commutativity of above diagram in terms of two matrix
equations, and a simple matrix computation shows that $\alpha_{1}=1$ and
$\alpha_{2}=1$ are the only solutions. This implies that every automorphism of
$G$ is central. Note that this also covers the case $p=2$ as the restrictions
of $f$ to $A/G^{\prime}$ and $B/G^{\prime}$ are linear.

For more detailed module-theoretic formulation of the above arguments for
arbitrary special $p$-groups, one may refer to [6].

(4.5) The third instance of examples of Miller groups, not in the context of
the topic, is by Heineken [19] in 1979. He constructed a family of $p$-groups
in which every normal subgroup is characteristic. The groups, actually,
possess more interesting properties, which force the groups actually become
Miller. The construction is as follows. Let $\mathbb{F}_{q}$ ($q=p^{n}$) be
the field of order $p^{n}$ and ${\rm U}(3,\mathbb{F}_{q})$ denote the group of
$3\times 3$ unitriangular matrices over $\mathbb{F}_{q}$. Identify the
elements of ${\rm U}(3,\mathbb{F}_{q})$ as triples over $\mathbb{F}_{q}$ by

$(x,y,z)\leftrightsquigarrow\begin{bmatrix}1&x&z\\\ &1&y\\\
&&1\end{bmatrix},\hskip 14.22636ptx,y,z\in\mathbb{F}_{q}.$

The sets $A=\\{(x,0,z):x,z\in\mathbb{F}_{q}\\}$ and
$B=\\{(0,y,z):y,z\in\mathbb{F}_{q}\\}$ constitute abelian subgroups of order
$q^{2}=p^{2n}$ and exponent $p$; they generate ${\rm U}(3,\mathbb{F}_{q})$ and
their intersection is the center of ${\rm U}(3,\mathbb{F}_{q})$. Also the
commutator of an element of $A$ with that of $B$ is given by

$[(x,0,z_{1}),(0,y,z_{2})]=(0,0,xy).$

Heineken constructed a group $G$ by a little modification in the commutator
and power relations of ${\rm U}(3,\mathbb{F}_{q})$ as follows. The group $G$
consists of triples $(x,y,z)$ over $\mathbb{F}_{q}$, in which

$A^{*}=\\{(x,0,z)\mid x,z\in\mathbb{F}_{q}\\}\mbox{ and }B^{*}=\\{(0,y,z)\mid
y,z\in\mathbb{F}_{q}\\}$

constitute abelian subgroups of order $q^{2}=p^{2n}$. The power relations are

$\displaystyle(x,0,z)^{p}$ $\displaystyle=(0,0,x^{p}-x),$
$\displaystyle(0,y,z)^{p}$ $\displaystyle=(0,0,y+y^{p}+\cdots+y^{p^{n-1}}).$

For a fixed generator $t$ of the multiplicative group of $\mathbb{F}_{q}$, the
commutator relations between $A^{*}$ and $B^{*}$ are defined as

$[(x,0,z_{1}),(0,y,z_{2})]=(0,0,xy-tx^{p}y^{p^{2}}).$

Then $G=\\{(x,y,z)\mid x,y,z\in\mathbb{F}_{q}\\}$ is a group, which is the
product of abelian subgroups $A^{*}$ and $B^{*}$, and $Z(G)=\\{(0,0,z)\mid
z\in\mathbb{F}_{q}\\}=A^{*}\cap B^{*}.$ Fix $x\neq 0$ and vary
$y\in\mathbb{F}_{q}$, then the elements $xy-tx^{p}y^{p^{2}}$ exhaust whole
$\mathbb{F}_{q}$. It follows that $G^{\prime}=Z(G)$, and therefore $G$ is a
special $p$-group of order $p^{3n}$. With the above setup, we have

######  Theorem 20.

Let $n$ be odd positive integer. If $n\geq 5$, $p>2$ or $n=3$, $p\geq 5$, then
every automorphism of $G$ is identity on $Z(G)$ as well as on $G/Z(G)$; hence
${\operatorname{Aut}}(G)$ is abelian.

The groups $G$ in the preceding theorem have the following remarkable
property: for every $x\in G-G^{\prime}$, the conjugacy class of $x$ in $G$ is
$xG^{\prime}$. The groups satisfying this property are called Camina groups.
The readers interested in Camina groups are referred to [11] and the
references therein.

An automorphism $\alpha$ of a group is said to be class-preserving if it takes
each element of the group to its conjugate. It is an easy exercise to show
that each automorphism of the group $G$ is class-preserving. This is not only
true for the groups $G$ considered above, but also for any Camina $p$-group
which is Miller as well.

######  Theorem 21.

Let $G$ be a $p$-group which is Miller as well as a Camina group. Then
automorphisms of $G$ are all class-preserving.

As a simple consequence, we have

######  Corollary 22.

Let $G$ be a $p$-group which is Miller as well as a Camina group. Then normal
subgroups of $G$ are all characteristic.

We remark that the examples of Jonah-Konvisser are not Camina groups and
extraspecial $p$-groups are Camina groups but not Miller. As the structure of
Camina $p$-groups of class $2$ in general is not well understood, it will be
interesting to study $p$-groups which are both Camina as well as Miller.

Problem 4. Determine the structure of $p$-groups which are Camina as well as
Miller.

This structural information will also, on the one hand, help in understanding
$p$-groups of class $2$ whose automorphisms are all class-preserving and, on
the other hand, shed some light on the study of $p$-groups of class $2$ in
which all normal subgroups are characteristic.

(4.6) Note that the Miller groups constructed by Heineken in the preceding
discussion are of order at least $p^{9}$. But, as we already mentioned, the
lower bound on the order of Miller groups is $p^{7}$ for odd $p$. This was
Morigi [37], who, in 1994, constructed examples of Miller groups of minimal
order as a part of a general construction of an infinite family of such
groups. The construction is briefly described as follows. For any natural
number $n$, let $G^{n}$ denote the $p$-group of class $2$ generated by
$a_{1},a_{2},b_{1},\ldots,b_{2n}$ with the following additional commutator and
power relations:

$\displaystyle[a_{1},b_{2i+1}]=[a_{2},b_{2i+2}]=1,\hskip
14.22636pti=0,1,\ldots,n-1,$
$\displaystyle[b_{1},b_{2}]=[b_{3},b_{4}]=\cdots=[b_{2n-1},b_{2n}]=1,$
$\displaystyle[b_{i},b_{j}]=1\mbox{ if }i\equiv j\pmod{2}.$

Further $a_{1}^{p}=a_{2}^{p}=1$ and $b_{1}^{p}$ is the product of following
$n^{2}+n+1$ commutators:

(**)
$[a_{1},a_{2}],[a_{1},b_{2i+2}],[a_{2},b_{2i+1}],[b_{2i+1},b_{2j+2}],\hskip
14.22636pti,j=0,1,\ldots,n-1,i\neq j.$

Finally powers of $b_{i}$’s are related by

$\displaystyle b_{2}^{p}=b_{1}^{p}[a_{1},b_{2}]^{-1}$ $\displaystyle
b_{2i+1}^{p}=b_{2i}^{p}[a_{2},b_{2i+1}]^{-1},\,\,\,\,b_{2i+2}^{p}=b_{2i+1}^{p}[a_{1},b_{2i+2}]^{-1},\hskip
14.22636pti=1,\ldots,n-1.$

Then $G^{n}$ is a special $p$-group, with $|G^{n}/(G^{n})^{\prime}|=p^{2n+2}$
and $(G^{n})^{\prime}$ is elementary abelian $p$-group generated by
$n^{2}+n+1$ commutators in $(**)$; so $|G^{n}|=p^{n^{2}+3n+3}$. Further,
${\operatorname{Aut}}(G^{n})$ is (elementary) abelian, and

$|{\operatorname{Aut}}(G^{n})|=|{\operatorname{Autcent}}(G^{n})|=p^{(2n+2)(n^{2}+n+1)}.$

For $n=1$, $G^{1}$ is a special $p$-group of order $p^{7}$. This is an example
of a Miller group of the smallest order for $p$ odd. The following problem is
interesting:

Problem 5. Describe all Miller $p$-groups of order $p^{7}$ for $p>2$.

(4.7) Upto this point, we have only considered the examples of special Miller
$p$-groups (modulo Remark 19). Now we’ll consider non-special groups. Before
we proceed further with more examples, we record a very useful result of Adney
and Yen [1]

Let $G$ be a purely non-abelian $p$-group of class $2$. Let
$G/G^{\prime}=\prod_{i=1}^{r}\langle x_{i}G^{\prime}\rangle$, with
$o(x_{i}G^{\prime})\geq o(x_{i+1}G^{\prime})$. Let $p^{a},p^{b},p^{c}$ denote
the exponents of $Z(G)$, $G^{\prime}$ and $G/G^{\prime}$ respectively.
Finally, let $R$ be the subgroup of $Z(G)$ generated by all the homomorphic
images of $G$ in $Z(G)$, and $K$ denote the intersection of the kernels of all
the homomorphisms $G\rightarrow G^{\prime}$.

######  Theorem 23 (Adney-Yen, [1]).

With the above notations, ${\operatorname{Autcent}}(G)$ is abelian if and only
if the following hold:

  1. (1)

$R=K$.

  2. (2)

either min$(a,c)=b$ or min$(a,c)>b$ and $R/G^{\prime}=\langle
x^{p^{b}}G^{\prime}\rangle$.

Motivated by the conjecture of Mahalanobis as stated in the introduction, Jain
and the second author [27], in 2012, constructed the following infinite family
of non-special Miller $p$-groups. For $n\geq 2$, and $p$ odd, let
$G_{n}=\langle x_{1},x_{2},x_{3},x_{4}\rangle$ be the $p$-group of class $2$
with the following additional relations:

$\displaystyle x_{1}^{p^{n}}=x_{2}^{p^{2}}=x_{3}^{p^{2}}=x_{4}^{p}=1,$
$\displaystyle[x_{1},x_{2}]=x_{2}^{p},\;[x_{1},x_{3}]=[x_{1},x_{4}]=x_{3}^{p},$
$\displaystyle[x_{2},x_{3}]=x_{1}^{p^{n-1}},\;[x_{2},x_{4}]=x_{2}^{p},\;[x_{3},x_{4}]=1.$

It is then easy to see that

  1. (1)

$Z(G_{n})=\Phi(G_{n})=\langle x_{1}^{p},x_{2}^{p},x_{3}^{p}\rangle$.

  2. (2)

$G_{n}^{\prime}=\langle x_{1}^{p^{n-1}},x_{2}^{p},x_{3}^{p}\rangle$ is
elementary abelian of order $p^{3}$.

  3. (3)

$G_{n}$ is special only when $n=2$ (follows by (1) and (2)).

The proof that $\operatorname{Aut}(G_{n})=\operatorname{Autcent}(G_{n})$ is
constructive and reply on detailed careful calculations. An application of
Theorem 23 now shows that $G_{n}$ is Miller.

In the preceding examples of non-special Miller $p$-groups $G$, one can easily
notice that

$G^{\prime}<\operatorname{Z}(G)=\Phi(G).$

One might desire that for a Miller $p$-group $G$, one of the following always
holds true: (i) $G^{\prime}=\operatorname{Z}(G)$; (ii)
$\operatorname{Z}(G)=\Phi(G)$.

In 2013, Jain, Rai and the second author [28] constructed the following
infinite family of Miller $p$-groups $G$ such that $G^{\prime}<Z(G)<\Phi(G)$,
which we again denote by $G_{n}$. For $n\geq 4$, let $G_{n}=\langle
x_{1},x_{2},x_{3},x_{4}\rangle$ be a $p$-group of class $2$ with the following
additional relations:

$\displaystyle x_{1}^{p^{n}}=x_{2}^{p^{4}}=x_{3}^{p^{4}}=x_{4}^{p^{2}}=1$
$\displaystyle[x_{1},x_{2}]=[x_{1},x_{3}]=x_{2}^{p^{2}},\;[x_{1},x_{4}]=x_{3}^{p^{2}},$
$\displaystyle[x_{2},x_{3}]=x_{1}^{p^{n-2}},\;[x_{2},x_{4}]=x_{3}^{p^{2}},\;[x_{3},x_{4}]=x_{2}^{p^{2}}.$

Then $G_{n}$ is a $p$-group of order $p^{n+10}$ with

$\displaystyle Z(G_{n})=\langle
x_{1}^{p^{2}},x_{2}^{p^{2}},x_{3}^{p^{2}}\rangle,\;\Phi(G_{n})=\langle
x_{1}^{p},x_{2}^{p},x_{3}^{p},x_{4}^{p}\rangle,\;G_{n}^{\prime}=\langle
x_{1}^{p^{n-2}},x_{2}^{p^{2}},x_{3}^{p^{2}}\rangle.$

It follows that $G_{n}^{\prime}\leq Z(G_{n})<\Phi(G_{n})$ and
$G_{n}^{\prime}=Z(G_{n})$ only when $n=4$. As in the previous examples, the
proof of the fact that $G_{n}$ is Miller is constructive.

As mentioned in Theorem 13, if ${\rm Aut}(G)$ is elementary abelian, then
$\Phi(G)$ is elementary abelian and one of the following holds: (1)
$Z(G)=\Phi(G)$; (2) $G^{\prime}=\Phi(G)$. Jain, Rai and the second author
constructed the following Miller $p$-groups in which exactly one of these
conditions hold.

For any prime $p$, let $G_{4}=\langle x_{1},x_{2},x_{3},x_{4}\rangle$ denote
the $p$-group of class $2$ with the following additional relations:

$\displaystyle x_{1}^{p^{2}}=x_{2}^{p^{2}}=x_{3}^{p^{2}}=x_{4}^{p^{2}}=1,$
$\displaystyle[x_{1},x_{2}]=1,\,[x_{1},x_{3}]=x_{4}^{p},\,[x_{1},x_{4}]=x_{4}^{p},$
$\displaystyle[x_{2},x_{3}]=x_{1}^{p},\,[x_{2},x_{4}]=x_{2}^{p},\,[x_{3},x_{4}]=x_{4}^{p}.$

Then $G_{4}$ is a $p$-group of order $p^{8}$ and the following holds.

  1. (1)

$G_{4}^{\prime}=\langle x_{1}^{p},x_{2}^{p},x_{4}^{p}\rangle$ is elementary
abelian of order $p^{3}$.

  2. (2)

$\Phi(G_{4})=Z(G_{4})=\langle x_{1}^{p},x_{2}^{p},x_{3}^{p},x_{4}^{p}\rangle$
is elementary abelian of order $p^{4}$.

  3. (3)

${\operatorname{Aut}}(G_{4})$ is elementary abelian of order $p^{16}$.

Again for any prime $p$, consider the $p$-group $G_{5}=\langle
x_{1},x_{2},x_{3},x_{4},x_{5}\rangle$ of class $2$ with the following
additional relations:

$\displaystyle
x_{1}^{p^{2}}=x_{2}^{p^{2}}=x_{3}^{p^{2}}=x_{4}^{p^{2}}=x_{5}^{p}=1,$
$\displaystyle[x_{1},x_{2}]=x_{1}^{p},\,[x_{1},x_{3}]=x_{3}^{p},\,[x_{1},x_{4}]=1,\,[x_{1},x_{5}]=x_{1}^{p},\,[x_{2},x_{3}]=x_{2}^{p},$
$\displaystyle[x_{2},x_{4}]=1,\,[x_{2},x_{5}]=x_{4}^{p},\,[x_{3},x_{4}]=1,\,[x_{3},x_{5}]=x_{4}^{p},\,[x_{4},x_{5}]=1.$

Then $G_{5}$ is a $p$-group of order $p^{9}$ and the following holds.

  1. (1)

$G_{5}^{\prime}=\Phi(G_{5})=\langle
x_{1}^{p},x_{2}^{p},x_{3}^{p},x_{4}^{p}\rangle$ is elementary abelian of order
$p^{4}$.

  2. (2)

$Z(G_{5})=\langle x_{4},G_{5}^{\prime}\rangle$.

  3. (3)

${\operatorname{Aut}}(G_{5})$ is elementary abelian of order $p^{20}$.

It is natural to ask

Question 6. Does there exist a Miller $p$-group $G$ in which
$Z(G)\nsubseteq\Phi(G)$ and $\Phi(G)\nsubseteq Z(G)$?

(4.8) The proofs of the results in subsection (4.7) involve heavy
computations. To remedy the problem, in 2015, Caranti [6, 7] suggested a
simple module theoretic approach to construct non-special Miller $p$-groups
from special ones. The arguments given in [6] are not sufficient to prove the
results as stated. The authors of the present survey proved that the results
are valid under an additional hypothesis. The construction is briefly
described as follows.

For an odd prime $p$, let $H$ be a special Miller $p$-group satisfying the
following hypotheses:

  1. (i)

$\mho_{1}(H)$ is a proper subgroup of $H^{\prime}$.

  2. (ii)

The map $H/H^{\prime}\rightarrow H^{\prime}$ defined by $hH^{\prime}\mapsto
h^{p}$ is injective.

Let $K=\left<z\right>$ be the cyclic group of order $p^{2}$, and $M$ be a
subgroup of order $p$ in $H^{\prime}$ but not in $\mho_{1}(H)$. Let $G$ be a
central product of $H$ and $K$ amalgamated at $M$. Note that
$G^{\prime}=\Phi(G)<Z(G)$; hence $G$ is non-special. With this setting, we
have

######  Theorem 24.

If $H/M$ is a Miller group, then so is $G$.

Before we proceed, we make a comment on the preceding theorem. Caranti claimed
that $G$ is Miller without the condition ‘$H/M$ is Miller’. Unfortunately,
this is not always true, as shown in the following example.

Let $H=\langle a,b,c,d\rangle$ be the $p$-group of class $2$ with the
following additional relations:

$a^{p}=[a,c],\hskip 8.53581ptb^{p}=[a,bcd],\hskip 8.53581ptc^{p}=[b,cd],\hskip
8.53581ptd^{p}=[b,d].$

Then $H$ is a special Miller $p$-group of order $p^{10}$ and satisfies
conditions (i)-(ii). It can be proved that if $M=\left<[a,b]\right>$, then $G$
is Miller, and if $M=\left<[a,d]\right>$, then $G$ is not a Miller group. That
$H$ and $H/\left<[a,b]\right>$ are Miller can be proved following the
arguments similar to those in [29] (for details see [31]).

As noted above, the Miller groups $G$ are such that $G^{\prime}=\Phi(G)<Z(G)$.
Now with a little variation in the preceding construction, we obtain Miller
$p$-groups $G$ with $G^{\prime}<\Phi(G)=Z(G)$. Assume that the special Miller
group $H$ also satisfies the following hypothesis in addition to (i)-(ii)
above:

  1. (iii)

If $H$ is minimally generated by $x_{1},\ldots,x_{n}$, then $H^{\prime}$ is
minimally generated by $[x_{i},x_{j}]$ for $1\leq i<j\leq n$.

Let $L=\left<z\right>$ be a cyclic group of order $p^{n}$, $n\geq 3$ and let
$z$ act on $H$ via a non-inner central automorphism $\sigma$ of $H$ (which
always exists in a Miller $p$-group). Let $N$ be a subgroup of order $p$ in
$H^{\prime}$ but not in $\mho_{1}(H)$. Now define $G$ to be the partial semi-
direct product of $H$ by $L$ amalgamated at $N$ (cf. [17] or [31]). With this
setting, we finally have

######  Theorem 25.

If $H/N$ is a Miller group, then so is $G$.

Again, we remark that the condition ‘$H/N$ is Miller’, in the preceding
theorem, can not be dropped, as shown by the following example. Let $H=\langle
a,b,c,d\rangle$ be the $p$-group of class $2$ described above. Then $\sigma$
defined by

$a\mapsto ad^{p},\hskip 14.22636ptb\mapsto b,\hskip 14.22636ptc\mapsto
c,\hskip 14.22636ptd\mapsto d$

extends to a non-inner central automorphism of $H$. Let $L$ be of order
$p^{3}$ acting on $H$ via $\sigma$. Then $G$ is Miller if
$N=\left<[a,b]\right>$, and $G$ is not if $N=\left<[a,d]\right>$.

######  Remark 26.

The above construction of non-special Miller groups $G$ from special Miller
groups $H$ is valid even without hypotheses (i)-(iii) on $H$. That
${\operatorname{Aut}}(G)={\operatorname{Autcent}}(G)$ can be proved using the
same arguments as in [31], which do not rely on hypotheses (i)-(iii). Then one
can apply Theorem 23 to show that $\operatorname{Aut}(G)$ is abelian.

We conclude with the following remarks. In all known examples of Miller
$p$-groups $G$, $G/Z(G)$ is homocyclic. It will be interesting to know whether
this happens in all Millers $p$-groups. If true, one might expect if
$\gamma_{2}(G)$ is always homocyclic. Again in the known Millers $p$-groups
$G$ one can observe that either $G^{\prime}$ and $Z(G)$ have same ranks or
$G/Z(G)$ and $G/G^{\prime}$ have same ranks. We wonder whether this is true
for all Miller $p$-groups.

## References

  * [1] J. E. Adney and T. Yen, Automorphisms of $p$-group, Illinois J. Math. 9 (1965), 137-143.
  * [2] G. Ban and S. Yu, Minimal abelian groups that are not automorphism groups, Arch. Math. 70 (1998), 427-434.
  * [3] Y. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 2, Walter de Gruyter, Berlin, New York (2008).
  * [4] J. N. S. Bidwell, M. J. Curran and D. J. McCaughan, Automorphisms of direct products of finite groups, Arch. Math. 86 (2006), 481-489.
  * [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I: The user language, J. Symbolic Comput. 24 (1997), 235-265.
  * [6] A. Caranti, A Module theoretic approach to abelian automorphism groups, Israel J. Math. 205 (2015), 235-246.
  * [7] A. Caranti, Erratum to “A module-theoretic approach to abelian automorphism groups”, Israel J. Math. 215 (2016), 1025-1026.
  * [8] M. J. Curran, Semi-direct product groups with abelian automorphism groups, J. Austral. Math. Soc. (Series A) 42 (1987), 84-91.
  * [9] M. J. Curran and D. J. McChaughan, Central automorphisms that are almost inner, Comm. Algebra 29 (2001), 2081-2087.
  * [10] M. J. Curran, Direct products with abelian automorphism groups. Comm. Algebra 35 (2007), 389-397.
  * [11] R. Dark and C. M. Scoppola, On Camina groups of prime power order, J. Algebra 181 (1996), 787 - 802.
  * [12] B. E. Earnley, On finite groups whose group of automorphisms is abelian, Ph.D. Thesis, Wayne State University (1975).
  * [13] R. Faudree, A note on the automorphism group of a p-group, Proc. Amer. Math. Soc. 19 (1968), 1379 -1382.
  * [14] R. Faudree, Groups in which each element commutes with its epimorphic images, Proc. Amer. Math. Soc. 27, No. 2 (1971), 236-240.
  * [15] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.3, 2016 (http://www.gap-system.org).
  * [16] S. P. Glasby, $2$-groups with every automorphisms central, J. Austral. Math. Soc. (Series A), 41 (1986), 233-236.
  * [17] D. Gorenstein, Finite Groups, AMS Chelsea Pub., Second Ed. (1980).
  * [18] P. V. Hegarty, Minimal abelian automorphism groups of finite groups, Rend. Sem. Mat. Univ. Padova 94 (1995), 121-135.
  * [19] H. Heineken, Nilpotente Gruppen, deren sm̈atliche Normalteiler charakteristisch sind, Arch. Math. 33 (1979/80), 497 - 503.
  * [20] H. Heineken and M. Liebeck, The occurrence of finite groups in the automorphism group of nilpotent groups of class $2$, Arch. Math. 25 (1974), 8-16.
  * [21] G. T. Helleloid, A survey on automorphism groups of finite groups, arXiv:math/0610294v2 [math.GR], 25 Oct. 2006.
  * [22] H. Hilton, An Introduction to the Theory of Groups of Finite Order, Oxford at the Clarendon Press, 1908.
  * [23] C. Hopkins, Non-abelian groups whose groups of isomorphisms are abelian, Ann. Math. 29 (1) (1927), 508-520.
  * [24] A. Hughes, Automorphisms of nilpotent groups and supersolvable groups, Proc. Symp. Pure Math. 37 (1980), AMS, 205-207.
  * [25] M. H. Jafari, Elementary abelian $p$-groups as central automorphism groups, Comm. Algebra 34 (2006), 601-607.
  * [26] A. Jamali, Some new non-abelian $2$-groups with abelian automorphism groups, J. Group Theory 5 (2002), 53-57.
  * [27] V. K. Jain and M. K. Yadav, On finite $p$-groups whose automorphisms are all central, Israel J. Math. 189 (2012), 225-236.
  * [28] V. K. Jain, P. K. Rai and M. K. Yadav, On finite $p$-groups with abelian automorphism groups, Internat. J. Algebra Comput. 23 (2013), 1063-1077.
  * [29] D. Jonah and M. Konvisser, Some non-abelian $p$-groups with abelian automorphism groups, Arch. Math. 26 (1975), 131-133.
  * [30] T. Karimi and Z. K. Farimani, $p$-groups with elementary abelian automorphism groups, World Applied Programming, 1 (2011), 352 -354.
  * [31] R. D. Kitture and M. K. Yadav, Note on Caranti’s method of construction of Miller groups, Monatsh. Math. (to appear) DOI: 10.1007/s00605-016-0976-z.
  * [32] I. D. Macdonald, The Theory of Groups, Oxford at the Clarendon Press, 1968.
  * [33] A. Mahalanobis, Diffe-Hellman key exchange protocol and non-Abelian nilpotent groups, Israel J. Math. 165 (2008), 161-187.
  * [34] I. Malinowska, $p$-automorphisms of finite $p$-groups - problems and questions, Advances in Group Theory, Aracne Edritice, Rome (2002), 111-127.
  * [35] A. Mann, Some questions about $p$-groups, J. Austral. Math. Soc. (Series A) 67 (1999), 356-379.
  * [36] G. A. Miller, A non-abelian group whose group of isomorphisms is abelian, The Messenger of Mathematics XVIII (1913-1914), 124-125.
  * [37] M. Morigi, On $p$-groups with abelian automorphism group, Rend. Sem. Mat. Univ. Padova 92 (1994), 47-58.
  * [38] M. Morigi, On the minimal number of generators of finite non-abelian $p$-groups having an abelian automorphism group, Comm. Algebra 23 (1995), 2045-2065.
  * [39] R. R. Struik, Some non-abelian $2$-groups with abelian automorphism groups, Arch. Math. 39 (1982), 299-302.
  * [40] H. Zassenhaus, The Theory of Groups, Chelsea Publishing Company, New York, 1949.

Rahul Dattatraya Kitture,

Post Doctoral Fellow, School of Mathematics, Harish-Chandra Research
Institute, Chhatnag Road, Jhunsi, Allahabad 211019, INDIA & Homi Bhabha
National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085,
India E-mail address<EMAIL_ADDRESS>
Manoj K. Yadav,

School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road,
Jhunsi, Allahabad 211019, INDIA & Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400085, India E-mail address:
<EMAIL_ADDRESS>