The automorphism tower problem. II (Q1279629)

If the centre of the group \(G\) is trivial, then \(G\) is naturally embedded in \(\Aut(G)\) via the group of inner automorphisms which is, in this case, isomorphic to \(G\). Again the group \(\Aut(G)\) has trivial centre and thus can be embedded in its automorphism group. So there is a tower beginning with \(G_0=G\) and if \(G_\alpha\) is defined for some ordinal \(\alpha\), then \(G_{\alpha+1}\) is the automorphism group of \(G_\alpha\), containing \(G_\alpha\) via the above embedding. If \(\lambda\) is a limit ordinal then \(G_\lambda=\bigcup_{\alpha<\lambda} G_\alpha\). This is the automorphism tower. In a previous paper [Part I, Proc. Am. Math. Soc. 95, 166-168 (1985; Zbl 0575.20030)], the author has shown that the automorphism tower of a centreless group \(G\) terminates, i.e. there exists an ordinal \(\alpha\), such that \(G_\alpha=G_\beta\) for all ordinals \(\beta>\alpha\). In the same paper, he has shown that for each ordinal \(\alpha\), a group can be constructed whose automorphism tower terminates in exactly \(\alpha\) steps. In this paper, the author proves that the automorphism tower of an infinite centreless group \(G\) of cardinality \(\kappa\) terminates in less than \((2^\kappa)^+\) steps. If \(G\) is a centreless group then \(\tau(G)\) denotes the least ordinal \(\alpha\) such that \(G_\alpha=G_\beta\) for all \(\beta>\alpha\). So the main result of the paper states: It is consistent with ZFC that for all regular cardinals \(\kappa\), (a) \(2^\kappa>\kappa^{++}\); and (b) whenever \(G\) is a centreless group of cardinality \(\kappa\), then \(\tau(G)<\kappa^{++}\).