Robinson, Derek J. S. A contribution to the theory of groups with finitely many automorphisms. (English) Zbl 0358.20052 Proc. Lond. Math. Soc., III. Ser. 35, 34-54 (1977). 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. Reviewer: Derek J. S. Robinson (Urbana) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 13 Documents MSC: 20E36 Automorphisms of infinite groups PDFBibTeX XMLCite \textit{D. J. S. Robinson}, Proc. Lond. Math. Soc. (3) 35, 34--54 (1977; Zbl 0358.20052) Full Text: DOI