Some properties of central kernel and central autocommutator subgroups (Q2816891)

If \(G\) is a group, the subgroup \(K(G)\) consisting of all elements \(x\) such that \(x^\alpha=x\) for every central automorphism \(\alpha\) of \(G\) is called the central kernel of \(G\). The central kernel was introduced by \textit{F. Haimo} [Trans. Am. Math. Soc. 78, 150--167 (1955; Zbl 0064.02203)], and recently \textit{F. Catino} et al. [J. Algebra 409, 1--10 (2014; Zbl 1322.20022)] have carefully investigated the structure of groups in which the central kernel has finite index. In the paper under review, the authors prove that if \(G\) is a group such that \(G/K(G)\) is finite, then~\(G\) admits only finitely many central automorphisms, and the number of these automorphisms can be suitably bounded.