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.
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1 Introduction
Let G be a group. If g is an element of G and \(\alpha \) is an automorphism of G, the element
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
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.
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
because \(M\le L(G)\). Then
As the autocommutator \([g_h,\alpha ]\) belongs to M, the above equality allows us to define a new map
by putting
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,
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
Therefore \(f_\alpha \) is a homomorphism.
Let k be an element of the kernel of \(f_\alpha \). Then
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
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
Therefore \(f_\alpha f_\beta =f_{\alpha \beta }\), and hence the map
is a group homomorphism.
Consider now the inverse isomorphisms
and
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
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
can be considered. As \(G'=L(G)\le Z(G)\), we have
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
Therefore the map
is a homomorphism, which is promptly seen to be injective.
Conversely, if f is any homomorphism of \(G/G'\) into \(G'\), consider the map
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
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
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.
References
Hegarty, P.V.: The absolute centre of a group. J. Algebra 169, 929–935 (1994)
Moghaddam, M.R.R., Safa, H.: Some properties of autocentral automorphisms of a group. Ricerche Mat. 59, 257–264 (2010)
Robinson, D.J.S.: A course in the theory of groups, 2nd edn. Springer, Berlin (1996)
Schur, I.: Neuer Beweis eines Satzes über endliche Gruppen. Sitzber. Akad. Wiss. Berlin, pp. 1013–1019 (1902)
Yadav, M.K.: On automorphisms of some finite \(p\)-groups. Proc. Ind. Acad. Sci. Math. Sci. 118, 1–11 (2008)
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Communicated by Salvatore Rionero.
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de Giovanni, F., Moghaddam, M.R.R. & Rostamyari, M.A. A note on autocentral automorphisms of groups. Ricerche mat. 64, 339–344 (2015). https://doi.org/10.1007/s11587-015-0242-z
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DOI: https://doi.org/10.1007/s11587-015-0242-z