More on core-2 2-groups
Let \(G\) be a group and \(H\) be a subgroup. Denote by \(H_G = \cap_{g \in G} H^g\) the core of \(H\). A finite group \(G\) is called a core-\(n\) group if \(|H : H_G| \leq n\) for all subgroups \(H\) of \(G\). This paper is devoted to \(2\)-groups \(G\), which are core-\(2\) groups. The main result says that any such group contains an abelian subgroup \(A\) of index at most 4, which contains \(\Phi (G)\). As a corollary one gets that any core-\(2\) group contains an abelian subgroup of index at most eight. This bound is best possible as \(G \cong D_{16} \ast Q_8\) shows. The corresponding result for odd \(p\), i.e., \(p\)-groups \(G\) (\(p\) odd) which are core-\(p\) have a normal abelian subgroup of index \(p^2\), was proved by \textit{G. Cutolo} et al. [J. Algebra 188, No. 2, 701--719 (1997; Zbl 0878.20014)].
scientific article; zbMATH DE number 6694617