Intersections of abelian subgroups in finite groups
Let \(G\) be a finite group with subgroups \(A\) and \(B\). The author of the paper under review calls minimal elements (with respect to inclusion) of the set \(\{A^g\cap B\mid g\in G\}\) minimal \((A, B)\)-intersections. Generalizing results of \textit{T. J. Laffey} [Proc. Edinb. Math. Soc., II. Ser. 20 (1976), 229-232 (1977; Zbl 0363.20021)], \textit{S. K. Wong} and the reviewer [ibid. 25, 19-20 (1982; Zbl 0479.20013)]\ he shows: Theorem 1. If \(A\) and \(B\) are abelian, then \(A^g\cap B\subseteq F(G)\) for minimal \((A, B)\)-intersections. Theorem 2. If \(A\) and \(B\) are cyclic, then \(A^g\cap B\trianglelefteq F(G)\) for minimal \((A,B)\)-intersections.
scientific article; zbMATH DE number 830718