On intersections of subgroups in groups (Q1580383)

The authors investigate the structure of a group \(G\) in the situation when the intersections of some subgroup \(H\) with its conjugates behave in a certain well described manner. Among other results the authors show: (1) If \(a\) and \(b\) are nonconjugate involutions of a periodic group \(G\) having finite centralizers, then \(|G:C_G(u)|\leq|G_G(a)|\cdot|G_G(b)|<\infty\) for some involution \(u\). (2) Let \(G\) be a group having a subgroup \(H\) of order \(n\) such that \(H\cap H^x\neq 1\) for \(x\in G-N_G(H)\). Then there is an element \(h\in H-1\) such that \(|G:C_G(h)|\leq(n-1)^2\). (3) Let \(G\) be a finite transitive permutation group of degree \(n\) with a cyclic stabilizer of a point. Then \(|G|\leq n^2-n\). This work is based on methods used by \textit{B. H. Neumann} [in: J. Lond. Math. Soc. 29, 236-248 (1954; Zbl 0055.01604)].