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A contribution to the theory of groups with finitely many automorphisms. (English) Zbl 0358.20052
Let \(G\) be a group with centre \(C\) and central quotient group \(Q=G/C\). Denote by \(\triangle\) the cohomology class of the central extension \(C \rightarrow G \rightarrow Q\). Then there is an associated exact sequence: \( 0 \rightarrow \mathrm{Hom}(Q_{ab},C) \rightarrow \mathrm{Aut}\;G\rightarrow C_{\mathrm{Aut}\; C \times \mathrm{Aut}\; Q} (\triangle)\rightarrow 1\) which can be used effectively to study groups whose automorphism groups are finite. We quote some results. \(\mathrm{Aut}\;G\) is finite if and only if \(Q\), \(\mathrm{Hom}(Q_{ab},C)\) and \(C_{\mathrm{Aut}\; C}(\triangle)\) are finite. Necessary and sufficient conditions are obtained for an abelian group \(A\) to be the centre of a group with finite automorphism group. One such condition is that the torsion subgroup of \(A\) be finite, as has been pointed out by Nagrebeckiń≠. On the other hand \(\mathrm{Aut}\; A\) need not be finite. An example is constructed of a group \(G\) such that \(\mathrm{Aut}\;G\cong S_{4}\) but the centre of \(G\) has uncountably many automorphisms.

MSC:
20E36 Automorphisms of infinite groups
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