A note on autocentral automorphisms of groups

Abstract

The absolute centre L(G) of a group G is the subgroup of all elements fixed by every automorphism of G, and an automorphism of G is autocentral if it acts trivially on the factor group G / L(G). Autocentral automorphisms have been introduced by Moghaddam and Safa (Ricerche Mat 59:257–264, 2010). The aim of this paper is to obtain new informations on the behaviour of autocentral automorphisms of a group. We also consider the relations between the group of autocentral automorphisms and that of class preserving automorphisms of a group.

1 Introduction

Let G be a group. If g is an element of G and $\alpha$ is an automorphism of G, the element

$$\begin{aligned}{}[g,\alpha ]=g^{-1}g^\alpha =g^{-1}\alpha (g) \end{aligned}$$

is the autocommutator of g and $\alpha$. Of course, if $\alpha$ is the inner automorphism determined by an element a of G, then the autocommutator $[g,\alpha ]$ coincides with the ordinary commutator [g, a]. The set L(G), consisting of all elements g of G fixed by every automorphism of G is a central characteristic subgroup of G, which is called the absolute centre (or the autocentre) of G. Then

$$\begin{aligned} L(G)=\{ g\in G\;|\; [g,\alpha ]=1 \; \forall \alpha \in \hbox {Aut}(G)\}, \end{aligned}$$

where $\hbox {Aut}(G)$ is the group of all automorphisms of G. Moreover, the subgroup K(G) generated by all autocommutators $[g,\alpha ]$ (where $g\in G$ and $\alpha \in \hbox {Aut}(G)$) is the autocommutator subgroup of G. Observe that if in the above considerations the full automorphism group $\hbox {Aut}(G)$ is replaced by the group $\hbox {Inn}(G)$ of all inner automorphisms of G, then we obtain the usual definitions of centre and commutator subgroup.

The absolute centre and the autocommutator subgroup have been introduced by Hegarty [1], who proved in particular that if the absolute centre L(G) of a group G has finite index, then both the autocommutator subgroup K(G) and the automorphism group $\hbox {Aut}(G)$ are finite, a result that must be compared with the celebrated theorem of Schur [4] on the finiteness of the commutator subgroup of a central-by-finite group.

Let G be a group, and let N be a characteristic subgroup of G. We shall denote by $\hbox {Aut}_N(G)$ the normal subgroup of $\hbox {Aut}(G)$ consisting of all automorphisms of G inducing the identity on the factor group G / N, i.e.

$$\begin{aligned} \hbox {Aut}_N(G)=\{\alpha \in \hbox {Aut}(G)\;|\; [g,\alpha ]\in N \; \forall g\in G\}. \end{aligned}$$

In particular, if we choose $N=Z(G)$, then $\hbox {Aut}_{Z(G)}(G)$ is precisely the group $\hbox {Aut}_c(G)$ of all central automorphisms of G. If $N=L(G)$, the group $\hbox {Aut}_{L(G)}(G)$ will be denoted here by $\hbox {Aut}_L(G)$; the elements of $\hbox {Aut}_L(G)$ are called autocentral automorphisms of G (see also [2], where this concept has been introduced). Our main result on the group of autocentral automorphisms is the following.

Theorem

Let G and $\bar{G}$ be groups such that $L(G)\simeq L(\bar{G})$ and $G/L(G)\simeq \bar{G}/L(\bar{G})$. Then the groups of autocentral automorphisms $\hbox {Aut}_L(G)$ and $\hbox {Aut}_L(\bar{G})$ are isomorphic.

Recall also that an automorphism $\alpha$ of a group G is called class preserving if the image $x^\alpha$ belongs to the conjugacy class $x^G$, for all $x\in G$. The set $\hbox {Aut}^c(G)$ of all class preserving automorphisms of G is a normal subgroup of $\hbox {Aut}(G)$, which of course contains the inner automorphism group $\hbox {Inn}(G)$. Class preserving automorphisms were introduced by Yadav [5], who investigated conditions under which two (finite) groups have isomorphic class preserving automorphism groups. The relations between the groups $\hbox {Aut}_L(G)$ and $\hbox {Aut}^c(G)$ are considered in the last part of the paper.

Most of our notation is standard and can be found in [3].

2 Statements and proofs

Let G be a group. If x is an element of G and $\alpha$ is an automorphism of G, the autocommutator $[x,\alpha ]$ is of course an ordinary commutator in the holomorph group of G. Thus the following lemma is just an application of the usual commutator laws; it gives rules that can be used in the study of autocommutators.

Lemma 1

Let x and y be elements of a group G and $\alpha , \beta \in \hbox {Aut}(G)$. Then the following identities hold:

• (a) $[xy,\alpha ]=[x,\alpha ]^y[y,\alpha ]$;

• (b) $[x,\alpha ^{-1}]=([x,\alpha ]^{-1})^{\alpha ^{-1}}$;

• (c) $[x^{-1},\alpha ]=([x,\alpha ]^{-1})^{x^{-1}}$;

• (d) $[x,\alpha \beta ]=[x,\beta ][x,\alpha ]^\beta =[x,\beta ][x,\alpha ][x,\alpha ,\beta ]$;

• (e) $[x,\alpha ]^\beta =[x^\beta ,\alpha ^\beta ]$;

• (f) $[x,\alpha ^{-1},\beta ]^\alpha [\alpha ,\beta ^{-1},x]^\beta [\beta ,x^{-1},\alpha ]^x=1$.

Our main theorem is a special case of the following result.

Theorem 1

Let G and H be groups, and let M and N be characteristic subgroups of G and H, respectively, such that $M\le L(G)$ and $N\le L(H)$. If $G/M\simeq H/N$ and $M\simeq N$, then the groups $\hbox {Aut}_M(G)$ and $\hbox {Aut}_N(H)$ are isomorphic.

Proof

Let Let $\varphi : G/M\longrightarrow H/N$ and $\psi : M\longrightarrow N$ be isomorphisms. Let $\alpha$ be any element of the group $\hbox {Aut}_M(G)$. Clearly, for each element h of H there exists an element $g_h$ of G such that $\varphi (g_hM)=hN$ and $h\in N$ if and only if $g_h\in M$. If g is any other element of G such that $\varphi (gM)=hN$, the product $g_hg^{-1}$ belongs to M and hence

$$\begin{aligned} g_h^\alpha (g^{-1})^\alpha =(g_hg^{-1})^\alpha =g_hg^{-1}, \end{aligned}$$

because $M\le L(G)$. Then

$$\begin{aligned}{}[g,\alpha ]=g^{-1}g^\alpha =g_h^{-1}g_h^\alpha =[g_h,\alpha ]. \end{aligned}$$

As the autocommutator $[g_h,\alpha ]$ belongs to M, the above equality allows us to define a new map

$$\begin{aligned} f_\alpha : H\longrightarrow H \end{aligned}$$

by putting

$$\begin{aligned} f_\alpha (h)=h\psi ([g_h,\alpha ]) \end{aligned}$$

for each element h of H. Observe here that if h belongs to N, then $g_h$ lies in M, so that $[g_h,\alpha ]=1$ and hence $f_\alpha (h)=h$, i.e. the restriction of $f_\alpha$ to N is the identity map.

Let $h_1$ and $h_2$ be arbitrary elements of H. Clearly,

$$\begin{aligned} \varphi (g_{h_1}g_{h_2}M)=\varphi (g_{h_1}M)\varphi (g_{h_2}M)=h_1h_2N, \end{aligned}$$

and so $[g_{h_1}g_{h_2},\alpha ]=[g_{h_1h_2},\alpha ]$. Since M is contained in Z(G) and N lies in Z(H), it follows that

$$\begin{aligned} f_\alpha (h_1)f_\alpha (h_2)= & {} h_1\psi ([g_{h_1},\alpha ])h_2\psi ([g_{h_2},\alpha ])=h_1h_2\psi ([g_{h_1},\alpha ][g_{h_2},\alpha ])\\= & {} h_1h_2\psi ([g_{h_1}g_{h_2},\alpha ])=h_1h_2\psi ([g_{h_1h_2},\alpha ])=f_\alpha (h_1h_2). \end{aligned}$$

Therefore $f_\alpha$ is a homomorphism.

Let k be an element of the kernel of $f_\alpha$. Then

$$\begin{aligned} 1=f_\alpha (k)=h\psi ([g_k,\alpha ]), \end{aligned}$$

so that $k=\psi ([g_k,\alpha ])^{-1}$ belongs to N, and hence $k=f_\alpha (k)=1$. Therefore the homomorphism $f_\alpha$ is injective. Moreover, if h is any element of H, we have

$$\begin{aligned} f_\alpha \bigl (h\psi ([g_h,\alpha ]^{-1}\bigr )= & {} f_\alpha (h)\psi ([g_h,\alpha ]^{-1})\\= & {} h\psi ([g_h,\alpha ])\psi ([g_h,\alpha ]^{-1})=h. \end{aligned}$$

It follows that $f_\alpha$ is also surjective, and hence it is an automorphism of H. Observe also that $[h,f_\alpha ]=h^{-1}f_\alpha (h)=\psi ([g_h,\alpha ]$ belongs to N for each element h of H, and so the automorphism $f_\alpha$ belongs to the group $\hbox {Aut}_N(H)$.

Let $\alpha$ and $\beta$ be elements of $\hbox {Aut}_M(G)$, and let h be any element of H. Then

$$\begin{aligned} (f_\alpha f_\beta )(h)= & {} f_\beta \bigl (f_\alpha (h)\bigr )=f_\beta \bigl (h\psi ([g_h,\alpha ])\bigr )\\= & {} f_\beta (h)f_\beta \bigl (\psi ([g_h,\alpha ])\bigr )=h\psi ([g_h,\beta ])\psi ([g_h,\alpha ]) \\= & {} h\psi ([g_h,\beta ][g_h,\alpha ])=h\psi ([g_h,\beta ][g_h,\alpha ]^{\beta })\\= & {} h\psi ([g_h,\alpha \beta ])=f_{\alpha \beta }(h). \end{aligned}$$

Therefore $f_\alpha f_\beta =f_{\alpha \beta }$, and hence the map

$$\begin{aligned} \tau : \alpha \in \hbox {Aut}_M(G)\longmapsto f_\alpha \in \hbox {Aut}_N(H) \end{aligned}$$

is a group homomorphism.

Consider now the inverse isomorphisms

$$\begin{aligned} \varphi ^{-1} : H/N\longrightarrow G/M \end{aligned}$$

and

$$\begin{aligned} \psi ^{-1} : N\longrightarrow M. \end{aligned}$$

The above method applied to $\varphi ^{-1}$ and $\psi ^{-1}$ allows to construct, for each element $\gamma$ of $\hbox {Aut}_N(H)$, an automorphism $f_\gamma$ of G which belongs to $\hbox {Aut}_M(G)$, and the map

$$\begin{aligned} \omega : \gamma \in \hbox {Aut}_N(H)\longmapsto f_\gamma \in \hbox {Aut}_M(G) \end{aligned}$$

is a homomorphism. It is easy to prove that $\omega \circ \tau$ is the identity map of $\hbox {Aut}_M(G)$ and $\tau \circ \omega$ is the identity map of $\hbox {Aut}_N(H)$. Therefore $\tau$ is an isomorphism and $\hbox {Aut}_M(G)\simeq \hbox {Aut}_N(H)$. $\square$

It was remarked in the introduction that if a is any element of a group G and $\alpha$ is the inner automorphism of G determined by a, the autocommutator $[g,\alpha ]$ coincides with the ordinary commutator [g, a] for each element g of G. It follows that if the inner automorphism $\alpha$ is autocentral, then the subgroup [G, a] is contained in L(G), i.e. the coset aL(G) belongs to the centre of G / L(G). In particular, if the group $\hbox {Aut}^c(G)$ of all class preserving automorphisms is contained in $\hbox {Aut}_L(G)$, we obtain that the commutator subgroup $G'$ lies in the absolute centre L(G) of G.

Our second main result shows that if G is any finite group in which the commutator subgroup and the absolute centre coincide, then $\hbox {Aut}^c(G)=\hbox {Aut}_L(G)$.

Theorem 2

Let G be a finite group such that $G'=L(G)$. Then $\hbox {Aut}^c(G)\simeq \hbox {Hom}\bigl (G/G',G'\bigr )$ and $\hbox {Aut}^c(G)=\hbox {Aut}_L(G)$.

Proof

Let $\alpha$ be any class preserving automorphism of G. Clearly, $[xu,\alpha ]=[x,\alpha ]$ for all elements x of G and u of $G'=L(G)$, and so the map

$$\begin{aligned} f_\alpha : xG'\in G/G'\longmapsto [x,\alpha ]\in G' \end{aligned}$$

can be considered. As $G'=L(G)\le Z(G)$, we have

$$\begin{aligned} f_\alpha (xyG')=[xy,\alpha ]=[x,\alpha ][y,\alpha ]=f_\alpha (xG')f_\alpha (yG') \end{aligned}$$

for all elements x and y of G, and hence $f_\alpha$ is a homomorphism. Observe also that, if $\alpha$ and $\beta$ are two class preserving automorphisms of G, and x is any element of G, then

$$\begin{aligned} f_{\alpha \beta }(xG')=[x,\alpha \beta ]=[x,\beta ][x,\alpha ]^\beta =[x,\alpha ][x,\beta ]=(f_\alpha +f_\beta )(x). \end{aligned}$$

Therefore the map

$$\begin{aligned} \psi : \alpha \in \hbox {Aut}^c(G)\longmapsto f_\alpha \in \hbox {Hom}(G/G',G') \end{aligned}$$

is a homomorphism, which is promptly seen to be injective.

Conversely, if f is any homomorphism of $G/G'$ into $G'$, consider the map

$$\begin{aligned} \alpha _f : G\longrightarrow G, \end{aligned}$$

defined by putting $\alpha _f(x)=xf(xG')$ for each element x of G. It is clear that f is a homomorphism. If x is an element of G such that $\alpha _f(x)=1$, then $x=f(xG')^{-1}$ belongs to $G'$, and so $x=1$. Therefore $\alpha _f$ is injective, and hence it is an automorphism of the finite group G. Moreover, $\alpha _f$ acts trivially on $G/G'$, and so it is an autocentral automorphism of G, because $G'=L(G)$. Finally, we have

$$\begin{aligned} \psi (\alpha _f)=f_{\alpha _f}=f, \end{aligned}$$

so that $\psi$ is an isomorphism, and the groups $\hbox {Aut}^c(G)$ and $\hbox {Hom}(G/G',G')$ are isomorphic.

On the other hand, $\hbox {Aut}_L(G)$ is naturally isomorphic to the homomorphism group $\hbox {Hom}\bigl (G/L(G),L(G)\bigr )$ (see also [2], Proposition 1), and hence in our case we obtain

$$\begin{aligned} \hbox {Aut}^c(G)\simeq \hbox {Aut}_L(G). \end{aligned}$$

As $\hbox {Aut}^c(G)$ acts trivially on $G/G'$, and $G'=L(G)$, it follows that all class preserving automorphisms are autocentral, so that $\hbox {Aut}^c(G)=\hbox {Aut}_L(G)$, and the proof is complete. $\square$

Observe finally that part of the statement of Theorem 2 can be generalized to certain types of infinite groups. Recall that a group G is cohopfian if it not isomorphic to any of its proper subgroups, i.e. if every injective endomorphism of G is an automorphism; for instance, every Černikov group is obviously cohopfian. The argument of the above proof can be used to show that if G is any cohopfian group such that $G'=L(G)$, then the groups $\hbox {Aut}^c(G)$ and $\hbox {Aut}_L(G)$ are isomorphic.