On equality of derival and inner automorphisms of some p-groups

Abstract

For a group G, D(G) denotes the group of all derival automorphisms of G. For a finite nilpotent group of class 2, it is shown that $D(G)\cong Hom(G/{\mathit{\gamma }}_2(G),{\mathit{\gamma }}_2(G))$. We prove that if G is a nilpotent group of class $\ge 3$ such that $Z(G)\subseteq {\mathit{\gamma }}_2(G)$ and $D(G/Z(G))=Inn(G/Z(G))$, then $D(G)=Inn(G)$ if and only if $Autcent(G)=Z(Inn(G))$. Finally, for an odd prime p, we classify all p-groups of order $p^n,1\le n\le 5$, for which $D(G)=Inn(G)$.

Keywords:

• automorphism
• inner automorphism
• class preserving automorphism
• derival automorphism
• central automorphism
• p-groups
• nilpotent groups
• isoclinic groups

AMS Subject Classifications:

• 20C07
• 13C05

Public Interest Statement

Groups are the most basic algebraic structure which is the building block of modern algebra. Groups handle many practical problems like symmetries of objects and various problems of combinatorics. Automorphism group is one of the most fascinating object associated with a group. In recent past, many researchers proposed different automorphism groups and their equality has been established with each other. This article is one in this sequence. The notion of derival automorphism group is introduced and its equality with the groups of inner automorphisms and class-preserving automorphisms is discussed. Results obtained are very fundamental in nature and will be quite helpful for researchers working in group theory.

1. Introduction

Let G be a group. Notations used are standard, however for the sake of completeness, by e we denote the identity element of G. For $x,y\in G$, $x^y$ denotes the conjugate element $y^{-1}xy$ and $[x,y]=x^{-1}{y}^{-1}xy$ is the commutator of x and y. $x^G$ denotes the conjugacy class of x in G. The subgroup generated by the set of all commutators of G is called derived group of G and it is denoted by [G, G] or $G^{{}^{\prime }}$ or ${\mathit{\gamma }}_2(G)$. [x, G] denotes the set of all commutators [x, g] for $g\in G$. Note that $x^g=x[x,g]$ for every $g\in G$ and $x^G=x[x,G]$. An endomorphism $f:G⟶G$ is called a class preserving if for each $x\in G$, $f(x)\in x^G$. Note that if f is a class-preserving endomorphism of G, then $x^{-1}f(x)\in [x,G]$. By $Ho{m}_c(G)$, we denote the set $\{f\in End(G)|f(x)\in [x,G]foreachx\in G$} . An automorphism $f:G⟶G$ is called a class-preserving automorphism if for each $x\in G$, $f(x)\in x^G$. Note that the inner automorphism $T_a:G⟶G$ given by $T_a(x)=a^{-1}xa$, for all $x\in G$, is a particular example of a class-preserving automorphism. The group of all class-preserving automorphisms is denoted by $Au{t}_c(G)$. The group of all inner automorphisms of G is denoted by Inn(G) and it is a normal subgroup of $Au{t}_c(G)$. $Ou{t}_c(G)$ denotes the quotient group $Au{t}_c(G)/Inn(G)$. Let N be a characteristic subgroup of G. Then each $\mathit{\alpha }\in Aut(G)$, induces an automorphism $\overline{\mathit{\alpha }}:G/N\to G/N$ given by $\overline{\mathit{\alpha }}(xN)=\mathit{\alpha }(x)N$. Thus the map $\mathit{\theta }:Aut(G)\to Aut(G/N)$ given by $\mathit{\theta }(\mathit{\alpha })=\overline{\mathit{\alpha }}$ is a homomorphism of groups. The kernel of this homomorphism is precisely those automorphisms of G which are identity on G / N. If we take $N=G^{{}^{\prime }}$, then $Ker\mathit{\theta }$ is the group of all those automorphisms of G which are identity on $G/G^{{}^{\prime }}$. This group is abbreviated as D(G) and elements of this group are called derival automorphisms of G (Chiṣ, 2002). An automorphism of G is called central if it is identity on G / Z(G). The set of all central automorphisms of G is a normal subgroup of Aut(G) and it is denoted by Autcent(G). It has been shown by Sah (1968) that $Autcent(G)=C_{Aut(G)}(Inn(G))$. In the recent past, interest of many mathematicians turned on the equalities of various automorphism groups viz. equalities of Autcent(G) and Inn(G), Autcent(G) and Z(Inn(G)) and Autcent(G) and $Au{t}_c(G)$ etc. Curran and McCaughan (2001) showed that if G is a finite p-group, then $Autcent(G)=Inn(G)$ if and only if $G^{{}^{\prime }}=Z(G)$ and Z(G) is cyclic. Further extending this work, Curran (2004) observed that Autcent(G) is minimum possible when $Autcent(G)=Z(Inn(G))$ and he found that if $Autcent(G)=Z(Inn(G))$, then $Z(G)\subseteq G^{{}^{\prime }}$ and Z(Inn(G)) must be cyclic. Gumber and Sharma (2011) proved that if G is a nilpotent group of class 2, then $Autcent(G)=Z(Inn(G))$ if and only if $G^{{}^{\prime }}=Z(G)$ and Z(G) is cyclic. For a finite p-group, Jafari (2011) find out the necessary and sufficient condition when every central automorphism fix Z(G) element-wise. Jain (2012) studied those finite p-groups for which $Aut(G)=Autcent(G)$. Kalra and Gumber (2013) characterize all finite p-groups of order $p^n,1\le n\le 7$, such that $Au{t}_c(G)=Autcent(G)$. Further Yadav (2013) proved that if G is a finite p-group such that $Au{t}_c(G)=Autcent(G)$, then G has even number of elements in any minimal generating set for G. Ghoraishi (2015) find out the necessary and sufficient condition for equality of class preserving and central automorphism of a finite group.

Kumar and Vermani (2000,2001) show that if G is a group of order $p^n$, $1\le n\le 4$, then $Au{t}_c(G)=Inn(G)$. In another note Yadav (2008) studied class-preserving automorphisms of group of order $p^5$, p an odd prime and proved that $Au{t}_c(G)=Inn(G)$ for all groups G of order $p^5$ except two isoclinism families. On the classification for group of order $p^6$ given by James (1980), Narain and Karan (2014) studied those groups of order $p^6$ for which $Au{t}_c(G)=Inn(G)$.

If $\mathit{\varphi }$ is either an inner automorphism or a class-preserving automorphism of G, then one should note that for all $x\in G$, $x^{-1}\mathit{\varphi }(x)\in [x,G]\subseteq G^{{}^{\prime }}$, whereas if $\mathit{\varphi }$ is a derival automorphism of G, then for all $x\in G$, $x^{-1}\mathit{\varphi }(x)\in G^{{}^{\prime }}$. This shows that$$\begin{array}{c}\hfill Inn(G)\subseteq Au{t}_c(G)\subseteq D(G).\end{array}$$

This motivate us to study those p-groups for which D(G) coincides with $Au{t}_c(G)$ or Inn(G). This paper is an attempt to study some p-groups for which $D(G)=Au{t}_c(G)$. One quite natural situation when $D(G)=Au{t}_c(G)$ is that $D(G)=Inn(G)$. Since for a p-group of order $p^n$, $1\le n\le 4$, $Au{t}_c(G)=Inn(G)$, to establish the equality of D(G) with $Au{t}_c(G)$ it is sufficient to show that $D(G)=Inn(G)$. We characterize all finite p-groups of class 2 for which $D(G)=Inn(G)$. For an odd prime p, we also classify those groups of order $p^5$ for which $D(G)=Inn(G)$.

2. Preliminaries and definitions

This section deals with some of the basic definitions and results which are used further.

Definition 1

For a group G, the sequence ${\{Z_i\}}_{i\ge 0}$ of subgroups of G defined as follows$$\begin{array}{c}\hfill Z_0=\{1\}andfori>0,Z_i/Z_{i-1}=Z(G/Z_{i-1}),\end{array}$$

is called the upper central series of G ; its i-th term is called the i-th center of G. Here $Z_1=Z(G)$, the center of G. A group G is said to be nilpotent if $Z_m=G$, for some positive integer m. The smallest integer c such that $Z_c(G)=G$, is called the nilpotency class of G .

Definition 2

For a group G, the sequence ${\{{\mathit{\gamma }}_i(G)\}}_{i\ge 1}$ of subgroups of G, defined by ${\mathit{\gamma }}_1(G)=G$ and ${\mathit{\gamma }}_{i+1}(G)=[G,{\mathit{\gamma }}_i(G)]$, is called the lower central series of G. Here ${\mathit{\gamma }}_2(G)=G^{{}^{\prime }}$, the derived group of G.

A group G is nilpotent if ${\mathit{\gamma }}_m(G)=\{e\}$, for some positive integer m. The smallest integer c such that ${\mathit{\gamma }}_{c+1}(G)=\{e\}$, is called the nilpotency class of G.

Definition 3

A group G is called nilpotent group of class 2 if G has a lower central series of the form $G={\mathit{\gamma }}_1(G)\supseteq {\mathit{\gamma }}_2(G)\supseteq {\mathit{\gamma }}_3(G)=\{e\}$.

The quaternion group $Q_8$ is an example of a nilpotent group of class 2. In fact every non-abelian group of order, $p^3$ is a nilpotent group of class 2.

Definition 4

A group is called a Camina group if and only if $G^{{}^{\prime }}\subseteq [x,G]$ for each $x\in G-G^{{}^{\prime }}$.

Every abelian group is a Camina group trivially. $Q_8=<a,b|a^4=1,b^2=a^2,b^{-1}ab=a^{-1}>$, the quaternion group is a non-abelian p-group of order 8 with $|Z(Q_8)|$=p. Note that derived group of $Q_8$ coincides with the center and it is a non-abelian Camina p-group (for proof see Lemma 2.1).

Definition 5

Let G be a group in which each element is of finite order, then the exponent of G is the least common multiple of orders of all elements and it is denoted by exp(G) .

Following results are important for further study.

Lemma 2.1

Let G be a p-group of class 2. Then for each $x\in G$, [x, G] is a subgroup of G. Moreover if $|G^{{}^{\prime }}|=|Z(G)|=p$, then G is a Camina p-group.

Proof

Let G be a nilpotent group of class 2. Then $G^{{}^{\prime }}\subseteq Z(G)$. For $x\in G$, consider $H=[x,G]$.

Let $[x,g_1],[x,g_2]\in H$. Then $[x,g_1{g}_2]=[x,g_2]{[x,g_1]}^{g_2}$. Since $G^{{}^{\prime }}\subseteq Z(G)$, $[x,g_1{g}_2]=[x,g_1][x,g_2]$. This shows that $[x,g_1][x,g_2]\in H$. Also for $[x,g]\in H$, $e=[x,e]=[x,g{g}^{-1}]=[x,g^{-1}]{[x,g]}^{g^{-1}}$. This shows that ${[x,g]}^{-1}=[x,g^{-1}]\in H$. Hence [x, G] is a subgroup of G. Let $x\in G-G^{{}^{\prime }}$. Then [x, G] is a non-trivial subgroup of $G^{{}^{\prime }}$. Since $|G^{{}^{\prime }}|$ is a prime, then $G^{{}^{\prime }}=[x,G]$ and hence G is a Camina group. $\square$

Theorem 2.2

(Yadav, 2008) Let G be a finite nilpotent group of class 2.

Then $Au{t}_c(G)\cong Ho{m}_c(G/Z(G),{\mathit{\gamma }}_2(G))$, where $Ho{m}_c(G/Z(G),{\mathit{\gamma }}_2(G))$ is the group $\{f\in Hom(G/Z(G),{\mathit{\gamma }}_2(G)):f(xZ(G))\in [x,G]forallx\in G\}$

Theorem 2.3

(Yadav, 2008) Let G be a finite p-group of class 2 such that ${\mathit{\gamma }}_2(G)$ is cyclic. Then, $Ou{t}_c(G)=1$ i.e. $Au{t}_c(G)=Inn(G)$.

Lemma 2.4

(Curran & McCaughan, 2001) Let G be a nilpotent group of class 2. Let Z(G), G / Z(G) and $G^{{}^{\prime }}$ have ranks z, r and d respectively. Then

(i)	$|Hom(G/Z(G)),G^{{}^{\prime }})|\ge |G/Z(G)|p^{r(d-1)}$.
(ii)	$|Hom(G/Z(G)),Z(G))|\ge |G/Z(G)|p^{r(z-1)}$.

3. Nilpotent groups of class 2

If G is an abelian group, then derived group $G^{{}^{\prime }}$ for such a group is trivial and hence Inn(G), $Au{t}_c(G)$ and D(G) contain merely the identity automorphism. Since abelian groups are precisely nilpotent group of class 1, this motivate us to study D(G) for nilpotent groups of class 2.

Theorem 3.1

Let G be a finite nilpotent group of class 2. Then $D(G)\cong Hom(G/{\mathit{\gamma }}_2(G),{\mathit{\gamma }}_2(G)$.

Proof

Let G be a nilpotent group of class 2. Then for each $f\in D(G)$, the map ${\mathit{\theta }}_f:G⟶{\mathit{\gamma }}_2(G)(\subseteq Z(G))$ defined by ${\mathit{\theta }}_f(x)=x^{-1}f(x)$, is a homomorphism of groups. Since ${\mathit{\theta }}_f$ sends elements of ${\mathit{\gamma }}_2(G)$ to 1, it induces homomorphism $\overline{{\mathit{\theta }}_f}:G/{\mathit{\gamma }}_2(G)⟶{\mathit{\gamma }}_2(G)$ given by $\overline{{\mathit{\theta }}_f}(x{\mathit{\gamma }}_2(G))=x^{-1}f(x)$. Thus we have the map $\mathit{\alpha }:D(G)⟶Hom(G/{\mathit{\gamma }}_2(G),{\mathit{\gamma }}_2(G))$ given by $\mathit{\alpha }(f)=\overline{{\mathit{\theta }}_f}$. Let $f,g\in D(G)$ and $x\in G-{\mathit{\gamma }}_2(G)$. Then there exists $a\in {\mathit{\gamma }}_2(G)$ such that $g(x)=xa$. Since ${\mathit{\theta }}_f$ sends elements of ${\mathit{\gamma }}_2(G)$ to 1,$$\begin{array}{cc}\hfill \overline{{\mathit{\theta }}_{f\circ g}}(x{\mathit{\gamma }}_2(G))& =x^{-1}fg(x)\hfill \\ \hfill & =x^{-1}f(g(x))\hfill \\ \hfill & =x^{-1}f(xa)\hfill \\ \hfill & =x^{-1}f(x)f(a)\hfill \\ \hfill & =x^{-1}f(x)a{a}^{-1}f(a)\hfill \\ \hfill & =x^{-1}f(x)a{\mathit{\theta }}_f(a)\hfill \\ \hfill & =x^{-1}f(x)a\hfill \\ \hfill & =x^{-1}f(x)x^{-1}g(x)\hfill \\ \hfill & =\overline{{\mathit{\theta }}_f}(x{\mathit{\gamma }}_2(G))\overline{{\mathit{\theta }}_g}(x{\mathit{\gamma }}_2(G))\hfill \\ \hfill \end{array}$$

This shows that $\mathit{\alpha }$ is a homomorphism of groups.

Let $f\in Hom(G/{\mathit{\gamma }}_2(G),{\mathit{\gamma }}_2(G)$. Then $f(x{\mathit{\gamma }}_2(G))\in {\mathit{\gamma }}_2(G)$.

Define a map $\mathit{\varphi }:G⟶G$ by $\mathit{\varphi }(x)=xf(x{\mathit{\gamma }}_2(G))$. It is easy to see that $\mathit{\varphi }$ is an endomorphism of G. Now $x^{-1}\mathit{\varphi }(x)=f(x{\mathit{\gamma }}_2(G))\in {\mathit{\gamma }}_2(G)$. Let $x\in ker(\mathit{\varphi })$. Then there are only two possibilities that either $x\in G-{\mathit{\gamma }}_2(G)$ or $x\in {\mathit{\gamma }}_2(G)$. Note that if $x\in G-{\mathit{\gamma }}_2(G)$, then $f(x{\mathit{\gamma }}_2(G))\ne x^{-1}$, otherwise $x\in {\mathit{\gamma }}_2(G)$. Thus $x\in ker\mathit{\varphi }$ if and only if $x\in {\mathit{\gamma }}_2(G)$. But then $1=\mathit{\varphi }(x)=xf(x{\mathit{\gamma }}_2(G))=xf({\mathit{\gamma }}_2(G))=x$. This shows that $ker\mathit{\varphi }=\{1\}$ and hence $\mathit{\varphi }$ is a monomorphism. Since G is a finite group, $\mathit{\varphi }$ is an automorphism of G. Thus $\mathit{\varphi }\in D(G)$.

Now $\overline{{\mathit{\theta }}_{\mathit{\varphi }}}$$(x{\mathit{\gamma }}_2(G))=x^{-1}\mathit{\varphi }(x)=f(x{\mathit{\gamma }}_2(G))$. Thus $\overline{{\mathit{\theta }}_{\mathit{\varphi }}}=f$. Since $\mathit{\alpha }(\mathit{\varphi })=\overline{{\mathit{\theta }}_{\mathit{\varphi }}}=f$, $\mathit{\alpha }$ is an epimorphism. It is fairly easy to check that $Ker(\mathit{\alpha })=\{1\}$. Thus $\mathit{\alpha }$ is an isomorphism and hence $D(G)\cong Hom(G/{\mathit{\gamma }}_2(G),{\mathit{\gamma }}_2(G))$. $\square$

Lemma 3.2

Let G be a p-group of class 2. Then, the order of each non-trivial element xZ(G) in G / Z(G) is equal to the exponent of the subgroup [x, G]

Proof

Let $x\in G-Z(G)$ and exp$([x,G])=p^n$. If $|xZ(G)|=p^m$. Then $x^{p^m}\in Z(G)$ and hence $[x^{p^m},a]=1$ for all $a\in G$. But then ${[x,a]}^{p^m}=1$ and hence $p^n\le p^m$ i.e. $n\le m$. Since exp $[x,G]=p^n$, for each $a\in G$$1={[x,a]}^{p^n}=[x^{p^n},a]$. This shows that $x^{p^n}\in Z(G)$ and hence $|xZ(G)|\le p^n$ i.e. $m\le n$. Thus |xZ(G)| = exp [x, G].

It is well known that in a finite p-group of class 2, exp$({\mathit{\gamma }}_2(G))$= exp (G / Z(G)).

Let G be a finite p-group of order $p^n$. Let $\{x_1,x_2\cdots x_r\}$ be a minimal generating set for G. If $|{\mathit{\gamma }}_2(G)|=p^m$, then by Burnside basis theorem $r\le n-m$. The following remarkable theorem (Theorem 5.1, Yadav, 2007) is quite useful in our context. $\square$

Theorem 3.3

Let G be a finite p-group. If $|Au{t}_c(G)|=p^{m(n-m)}$, then G is either abelian p-group or a non-abelian Camina special p-group.

Theorem 3.4

Let G be a finite p-group of class 2 such that $Au{t}_c(G)=D(G)$. Then the following holds

(1)	$|Hom(G/G^{{}^{\prime }},G^{{}^{\prime }})|=|Ho{m}_c(G/Z(G),G^{{}^{\prime }})|=|Hom(G/Z(G),G^{{}^{\prime }})|$
(2)	If $\{x_1,x_2\cdots x_r\}$ be a minimal generating set

for G, then

(a)	$Hom(<\overline{x_i}>,G^{{}^{\prime }})=Hom(<\overline{x_i}>,[x_i,G])$ for each $\overline{x_i}=x_{i}Z(G)$,
(b)	$G^{{}^{\prime }}=[x_i,G]$, for each $i,1\le i\le r$.

Proof

 

(1)

Since for a finite p-group of class 2, $D(G)\cong Hom(G/G^{{}^{\prime }},G^{{}^{\prime }})$ and $Au{t}_c(G)\cong Ho{m}_c(G/Z(G),G^{{}^{\prime }})$. Thus $$\begin{array}{ccc}\hfill |Hom(G/Z(G),G^{{}^{\prime }})|& \ge \hfill & \hfill |Ho{m}_c(G/Z(G),G^{{}^{\prime }})|\\ \hfill & =|Hom(G/G^{{}^{\prime }},G^{{}^{\prime }})|\hfill \\ \hfill & \ge \hfill & \hfill |Hom(G/Z(G),G^{{}^{\prime }})|\end{array}$$ But then $|Hom(G/Z(G),G^{{}^{\prime }})|=|Ho{m}_c(G/Z(G),G^{{}^{\prime }})|=|Hom(G/G^{{}^{\prime }},G^{{}^{\prime }})|$.

(2)

Let $\{x_1,x_2\cdots x_r\}$ be a minimal generating set for G. Then $G/Z(G)=<\overline{x_1}>\times <\overline{x_2}>\times \cdots \times <\overline{x_r}>$.

(a)	Since $|Hom(G/Z(G),G^{{}^{\prime }})|=|Ho{m}_c(G/Z(G),G^{{}^{\prime }})|$, it follows that ${\prod }_{i=1}^r|Hom(<\overline{x_i}>,G^{{}^{\prime }})|={\prod }_{i=1}^r|Hom(<\overline{x_i}>,[x_i,G])|$. But then $|Hom(<\overline{x_i}>,G^{{}^{\prime }})|=|Hom(<\overline{x_i}>,[x_i,G])|$.
(b)	Since for a finite p-group of class 2, exp $({\mathit{\gamma }}_2(G))$ = exp (G / Z(G)), from part (a), it follows that for each i, $1\le i\le r$, $G^{{}^{\prime }}=[x_i,G]$.

$\square$

Theorem 3.5

Let G be a finite p-group of class 2. Then $Au{t}_c(G)=D(G)$ if and only if G is a Camina p-group.

Proof

If G is a Camina p-group, then $G^{{}^{\prime }}\subseteq [x,G]$ for all $x\in G-G^{{}^{\prime }}$. Let $f\in D(G)$. Then for each $x\in G$, $x^{-1}f(x)\in G^{{}^{\prime }}$. For $x\in G-G^{{}^{\prime }}$,$G^{{}^{\prime }}\subseteq [x,G]$, therefore $x^{-1}f(x)\in [x,G]$. If $x\in G^{{}^{\prime }}$, then $f(x)=x$. Thus we observe that for each $x\in G$, $x^{-1}f(x)\in G^{{}^{\prime }}\subseteq [x,G]$ and hence $D(G)\subseteq Au{t}_c(G)$.

Conversely suppose that $D(G)=Au{t}_c(G)$. Let $|G|=p^n$ and $|G^{{}^{\prime }}|=p^m$. Suppose $\{x_1,x_2\cdots x_r\}$ is a minimal generating set for G. Then $G/Z(G)=<\overline{x_1}>\times <\overline{x_2}>\times \cdots \times <\overline{x_r}>$. Now (from theorem 3.4 (1)), it follows that $|D(G)|=|Au{t}_c(G)|={\prod }_{i=1}^r|Hom(<\overline{x_i}>,[x_i,G])|$. Since $|<\overline{x_i}>|$ = exp $([x_i,G])$, $|Au{t}_c(G)|={\prod }_{i=1}^r|[x_i,G]|$. But for each i, $1\le i\le r$, $G^{{}^{\prime }}=[x_i,G]$, it follows that $|Au{t}_c(G)|=|G^{{}^{\prime }}{|}^r={(p^m)}^r={(p^m)}^{n-m}$. Thus by theorem 3.3, G is a Camina special p-group. $\square$

Remark

Since Camina special p-group is a particular kind of Camina groups, the above result holds good for Camina special p-groups.

Theorem 3.6

Let G be a p-group of class 2. Then $D(G)=Inn(G)$ if and only if G is a Camina group and ${\mathit{\gamma }}_2(G)$ is cyclic.

Proof

If $D(G)=Inn(G)$, then $D(G)\subseteq Au{t}_c(G)$ and hence G is a Camina group. Let G / Z(G) and ${\mathit{\gamma }}_2(G)$ have exponent $p^c$ and ranks r and d respectively. Since G is non-abelian, $r\ge 2$. It is well known that for a p-group, $|Hom(G/Z(G),{\mathit{\gamma }}_2(G))|\ge |G/Z(G)|.p^{r(d-1)}$. Now $|G/Z(G)|=|Inn(G)|=|D(G)|=|Hom(G/{\mathit{\gamma }}_2(G),{\mathit{\gamma }}_2(G))|\ge |Hom(G/Z(G),{\mathit{\gamma }}_2(G)|\ge |G/Z(G)|p^{r(d-1)}.$ Thus $p^{r(d-1)}\le 1$. But this is possible only when $d=1$. This shows that ${\mathit{\gamma }}_2(G)$ is cyclic.

Conversely suppose that G is a Camina group of class 2. Then $D(G)=Au{t}_c(G)$. Further if ${\mathit{\gamma }}_2(G)$ is cyclic then $Au{t}_c(G)=Inn(G)$ and hence $D(G)=Inn(G)$. $\square$

Theorem 3.7

Let G be a group such that $D(G)=Inn(G)$. If $Z(G)\subseteq {\mathit{\gamma }}_2(G)$, then $Autcent(G)=Z(Inn(G))$.

Proof

Let $f\in Autcent(G)$. Then for each $x\in G$, $x^{-1}f(x)\in Z(G)$. Thus $Autcent(G)\subseteq D(G)$. Since $D(G)=Inn(G)$, $Autcent(G)\subseteq Inn(G)$. But $Autcent(G)=C_{Aut(G)}(Inn(G))$, this shows that $Autcent(G)=Z(Inn(G))$. $\square$

Theorem 3.8

Let G be a finite p-group such that $Z(G)\subseteq {\mathit{\gamma }}_2(G)$ and $D(G/Z(G))=Inn(G/Z(G))$. Then $|D(G)|=|Autcent(G)||G/Z_2(G)|$. Moreover $D(G)=Inn(G)$ if and only if $Autcent(G)=Z(Inn(G))$.

Proof

Let G be a finite p-group. Since $Z(G)\subseteq {\mathit{\gamma }}_2(G)$, each $f\in D(G)$ induces a derival automorphism $\overline{f}$ on G / Z(G). Hence, we have a homomorphism $\mathit{\alpha }:D(G)⟶D(G/Z(G))$ given by $\mathit{\alpha }(f)=\overline{f}$. It is easy to see that $Ker(\mathit{\alpha })=D(G)\cap Autcent(G)$. Since $Z(G)\subseteq {\mathit{\gamma }}_2(G)$, $Autcent(G)\subseteq D(G)$ and hence $Ker(\mathit{\alpha })=Autcent(G)$. But then D(G) / Autcent(G) is isomorphic to a subgroup of D(G / Z(G)). If $\overline{h}\in D(G/Z(G))$, then there exists an inner automorphism $T_a\in Inn(G)\subseteq D(G)$ and $\mathit{\alpha }(T_a)=\overline{T_a}=\overline{h}$. Thus $\mathit{\alpha }$ is an epimorphism. Thus $D(G)/Autcent(G)\cong Inn(G/Z(G))$. Now$$\begin{array}{cc}\hfill |D(G)|& =|Autcent(G)||G/Z_2(G)|\hfill \\ \hfill & =|Z_2(G)/Z(G)||G/Z_2(G)|\hfill \\ \hfill & =|G/Z(G)|\hfill \\ \hfill & =|Inn(G)|.\hfill \end{array}$$

Hence $D(G)=Inn(G)$. Conversely, suppose that $D(G)=Inn(G)$. Since, $Z(G)\subseteq G^{{}^{\prime }}$, by theorem 3.7, $Autcent(G)=Z(Inn(G))$. $\square$

4. Classification of groups of order $p^n(1\le n\le 5)$

Abelian groups satisfies $D(G)=Inn(G)$ trivially as the derived group for these groups is trivial. For an odd prime p, we classify all those groups of order $p^n,3\le n\le 5$ for which $D(G)=Inn(G)$.

4.1. Groups of order $p^3$

Let G be a non-abelian group of order $p^3$. Then G is a group of class 2 with $|Z(G)|=p$. But then $|[x,G]|=|G^{{}^{\prime }}|=|Z(G)|$. Thus G is a Camina group with cyclic derived group and hence $D(G)=Inn(G)$. From the above discussion every group of order $p^3$ satisfies $D(G)=Inn(G)$.

4.2. Groups of order $p^4$

In next two sections, we study groups of order $p^4$ and $p^5$ (p is an odd prime), on the basis of the classification given by James (1980). This classification is given in terms of isoclinism families. We start with the following definition of isoclinism of groups, given by Hall (1940).

Let X be a finite group and $\overline{X}=X/Z(X)$. Then commutation in X gives a well-defined map ${\mathit{\alpha }}_X:\overline{X}\times \overline{X}\mapsto {\mathit{\gamma }}_2(X)$ such that ${\mathit{\alpha }}_X(xZ(X),yZ(X))=[x,y]$ for $(x,y)\in X\times X$. Two finite groups G and H are called isoclinic if there exist isomorphisms$$\begin{array}{cc}\hfill \mathit{\theta }& :G/Z(G)\to H/Z(H),\hfill \\ \hfill \mathit{\varphi }& :{\mathit{\gamma }}_2(G)\to {\mathit{\gamma }}_2(H),\hfill \end{array}$$

such that $\mathit{\varphi }[\mathit{\alpha },\mathit{\beta }]=[{\mathit{\alpha }}^{{}^{\prime }},{\mathit{\beta }}^{{}^{\prime }}]$, for all $\mathit{\alpha },\mathit{\beta }\in G$, where ${\mathit{\alpha }}^{{}^{\prime }}Z(H)=\mathit{\theta }(\mathit{\alpha }(Z(G))$ and ${\mathit{\beta }}^{{}^{\prime }}Z(H)=\mathit{\theta }(\mathit{\beta }Z(G))$. The resulting pair $(\mathit{\theta },\mathit{\varphi })$ is called an isoclinism of G onto H. Clearly isomorphic groups are isoclinic but isoclinic groups need not be isomorphic. For example, $Q_8$ and $D_8$ are isoclinic groups which are not isomorphic. If G and H are isoclinic groups, then ${\mathit{\gamma }}_i(G)\cong {\mathit{\gamma }}_i(H)$ and ${\mathit{\delta }}_i(G)\cong {\mathit{\delta }}_i(H)$, whereas it is not necessary that $Z(G)\cong Z(H)$ (Hall, 1940). But one may observe that if G and H are finite isoclinc groups of equal order, then $|Z(G)|=|Z(H)|$. Since our further classification depends on the size of ${\mathit{\gamma }}_2(G)$ and that of Z(G), it is sufficient to calculate |D(G)| for only one member from each isoclinism family.

According to James (1980), for an odd prime, there are three isoclinism families of groups of order $p^4$ viz. ${\mathit{\varphi }}_1$,${\mathit{\varphi }}_2$ and ${\mathit{\varphi }}_3$. The family ${\mathit{\varphi }}_1$ corresponds to the family of abelian groups and hence $D(G)=Inn(G)$ for each member of this family.

The family ${\mathit{\varphi }}_2$ consists of groups of class 2 such that $G^{{}^{\prime }}$ is a cyclic group of prime order. Thus again $D(G)=Inn(G)$ for each member of this family.

In family ${\mathit{\varphi }}_3$, each group is a nilpotent group of class 3. Thus if $G\in {\mathit{\varphi }}_3$, then G is a group of maximal class and hence $|Z(G)|=p$. Since G / Z(G) is a non-abelian group of order $p^3$, by Section 4.1, $D(G/Z(G))=Inn(G/Z(G))$. For a group of maximal class, $Autcent(G)\ne Z(Inn(G))$. Thus in view of theorem 3.8, $D(G)\ne Inn(G)$.

From the above discussion, we conclude that a group of order $p^4$ (p is odd prime) satisfies $D(G)=Inn(G)$ if and only if either G is abelian or it is a nilpotent group of class 2 with $G^{{}^{\prime }}$, a cyclic group of prime order.

4.3. Groups of order $p^5$

For an odd prime there are 10 isoclinism families $({\mathit{\varphi }}_1-{\mathit{\varphi }}_{10})$ of groups of order $p^5$. The family ${\mathit{\varphi }}_1$ consists of abelian groups and hence $D(G)=Inn(G)$ for every group G lying in this family.

Theorem 4.1

If G is a nilpotent group of class 2, then $D(G)=Inn(G)$ if and only if G lie in the isoclinism families ${\mathit{\varphi }}_2$ and ${\mathit{\varphi }}_5$. Moreover in these cases $D(G)=Au{t}_c(G)$ as well.

Proof

Let G be a group of order $p^5$. Then, according to the classification given by James (1980), G is nilpotent group of class 2 if $G\in {\mathit{\varphi }}_2$, ${\mathit{\varphi }}_4$ and ${\mathit{\varphi }}_5$.

If $G={\mathit{\varphi }}_2{(321)}_a$, then $G={\mathit{\varphi }}_2(22)\times (1)$ where ${\mathit{\varphi }}_2(22)=<\mathit{\alpha },{\mathit{\alpha }}_1,{\mathit{\alpha }}_2|[{\mathit{\alpha }}_1,\mathit{\alpha }]={\mathit{\alpha }}^p={\mathit{\alpha }}_2,{\mathit{\alpha }}_1^{p^2}={\mathit{\alpha }}_2^p=1>$. The group $H={\mathit{\varphi }}_2(22)$ is a group of order $p^4$ having nilpotency class 2 and derived group of prime order. But then G is also a group of nilpotency class 2 which has a derived group of prime order. Hence, by theorem 3.6,$D(G)=Inn(G)$.

Let $G\in {\mathit{\varphi }}_4$. Suppose $G={\mathit{\varphi }}_4(1^5)$ , then $G=<\mathit{\alpha },{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\beta }}_1,{\mathit{\beta }}_2>$ such that $[{\mathit{\alpha }}_i,\mathit{\alpha }]={\mathit{\beta }}_i$ and ${\mathit{\alpha }}^p={\mathit{\alpha }}_i^p={\mathit{\beta }}_i^p=1$ , for $1\le i\le 2$. Thus G is a nilpotent group of class 2. Now $|D(G)|=|Hom(G/G^{{}^{\prime }},G^{{}^{\prime }})|=|Hom((\overline{\mathit{\alpha }}\times \overline{{\mathit{\alpha }}_1}\times \overline{{\mathit{\alpha }}_2}),({\mathit{\beta }}_1\times {\mathit{\beta }}_2))|=p^6$. Thus $D(G)\ne Inn(G)$.

If $G={\mathit{\varphi }}_5(1^5)$, then $G=<{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3,{\mathit{\alpha }}_4,\mathit{\beta }>$ such that $[{\mathit{\alpha }}_1,{\mathit{\alpha }}_2]=[{\mathit{\alpha }}_3,{\mathit{\alpha }}_4]=\mathit{\beta }$ and ${\mathit{\alpha }}_i^p=1={\mathit{\beta }}^p$. Since G is a nilpotent group of class 2 and $|G^{{}^{\prime }}|=p$, $D(G)=Inn(G)=Au{t}_c(G)$$\square$

Theorem 4.2

If G is a group of class 3, then $D(G)=Inn(G)$ if and only if $G\in {\mathit{\varphi }}_8$.

Proof

There are four isoclinism families ${\mathit{\varphi }}_3$, ${\mathit{\varphi }}_6-{\mathit{\varphi }}_8$ in this category.

Let $G={\mathit{\varphi }}_3(1^5)$. Then $G={\mathit{\varphi }}_3(1^4)\times (1)$. ${\mathit{\varphi }}_3(1^4)$ is a group of order $p^4$ having center of order p and nilpotency class 3. Thus G is also a nilpotent group of class 3 with $|Z(G)|=p^2$. But then $|G/Z(G)|=p^3$ and hence $D(G/Z(G))=Inn(G/Z(G))$.

If $G={\mathit{\varphi }}_6(1^5)$, then $G=<{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,\mathit{\beta },{\mathit{\beta }}_1,{\mathit{\beta }}_2>$ such that ${\mathit{\alpha }}_i^p={\mathit{\beta }}^p={\mathit{\beta }}_i^p=1$ and $[{\mathit{\alpha }}_1,{\mathit{\alpha }}_2]=\mathit{\beta }$, $[\mathit{\beta },{\mathit{\alpha }}_i]={\mathit{\beta }}_i$. Clearly G is a nilpotent group of class 3 and $|Z(G)|=p^2$. Thus, $|G/Z(G)|=p^3$ and hence $D(G/Z(G))=Inn(G/Z(G))$.

If $G={\mathit{\varphi }}_8(32)$. Then $G=<{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,\mathit{\beta }>$ such that $[{\mathit{\alpha }}_1,{\mathit{\alpha }}_2]=\mathit{\beta }={\mathit{\alpha }}_1^p$ and ${\mathit{\beta }}_{p^2}={\mathit{\alpha }}_2^{p^2}=1$. G is a nilpotent group of class 3 with $|Z(G)|=p$. Hence, G / Z(G) is a group of order $p^4$ with nilpotency class 2. But then $D(G/Z(G))=Inn(G/Z(G))$.

If G is a group of order $p^5$, then $Autcent(G)=Z(Inn(G))$ if and only if G is isomorphic to ${\mathit{\varphi }}_8(32)$ (Theorem 4.1, Gumber, 2011). Hence using theorem 3.8 in all the above three cases discussed above, we have $D(G)=Inn(G)$ if and only if $G={\mathit{\varphi }}_8(32)$.

We now left only with isoclinism family ${\mathit{\varphi }}_7$. Let G be a group ${\mathit{\varphi }}_7(1^5)$ from the isoclinism family ${\mathit{\varphi }}_7$. Then G is a nilpotent group of class 3 with center of prime order. In this case, $|Au{t}_c(G)|=p^5$ and $Au{t}_c(G)\ne Inn(G)$ (Yadav, 2008). Thus $D(G)\ne Inn(G)$, otherwise $Au{t}_c(G)=Inn(G)$. $\square$

Theorem 4.3

If G is a group from the isoclinism families ${\mathit{\varphi }}_9$ or ${\mathit{\varphi }}_{10}$. Then $D(G)\ne Inn(G)$.

Proof

Let G be a group ${\mathit{\varphi }}_{10}(1^5)$ from the isoclinism family ${\mathit{\varphi }}_{10}$. Then G is a nilpotent group of class maximal class with center of prime order. Again in this case $|Au{t}_c(G)|=p^5$ and $Au{t}_c(G)\ne Inn(G)$ (Lemma 5.2, Yadav, 2008). Thus $D(G)\ne Inn(G)$.

If G is the group ${\mathit{\varphi }}_9(1^5)$, then G is a nilpotent group with class 4 and $|Z(G)|=p$. Thus $Z(G)\subseteq {\mathit{\gamma }}_2(G)$. Clearly G / Z(G) is a group of order $p^4$ having nilpotency class 3. Thus $D(G/Z(G))\ne Inn(G/Z(G))$. Since $Z(G)\subseteq {\mathit{\gamma }}_2(G)$, $D(G)\ne Inn(G)$, otherwise $D(G/Z(G))=Inn(G/Z(G))$. $\square$

5. Conclusion

On the basis of classification of groups of order $p^n$$1\le n\le 5$ (for an odd prime p ), it is proved that if $|G|=p^r$, $1\le r\le 3$, then $D(G)=Inn(G)$. If $|G|=p^4$ , then $D(G)=Inn(G)$ if and only if either G is abelian or it is a nilpotent group of class 2 with a cyclic derived group of order p and if $|G|=p^5$, then $D(G)=Inn(G)$, if and only if $G\in {\mathit{\varphi }}_1$, ${\mathit{\varphi }}_2$, ${\mathit{\varphi }}_5$ and ${\mathit{\varphi }}_8$. A necessary and sufficient condition for nilpotent groups of class 2 is also obtained when $D(G)=Inn(G)$.

Acknowledgements

We thank the reviewers for their useful comments and suggestions.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Shiv Narain

Shiv Narain is an assistant professor in Arya P.G. College Panipat affiliated to Kurukshetra University, Kurukshetra, Haryana, India. He received his PhD degree from Kurukshetra university, Kurukshetra. He has been teaching undergraduate and postgraduate courses in Mathematics since last eight years. His field of research is group theory.

Ram Karan

Ram Karan is a professor in the Department of Mathematics Kurukshetra University, Kurukshetra, Haryana, India. He got his PhD from Kurukshetra university, Kurukshetra. He has more than 30 years of teaching experience in various postgraduate courses in Mathematics. His area of research includes Group-Rings, Group theory, Coding theory, and Homological algebra. He has published more than 20 research papers in journals of national and international repute.