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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 Nagrebeckiĭ. 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:
 2e+37 Automorphisms of infinite groups
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