Finite groups with X-quasipermutable Sylow subgroups

Abstract

Let H ≤ E and X be subgroups of a finite group G. Then we say that H is X-quasipermutable (XS-quasipermutable, respectively) in E provided that G has a subgroup B such that E = N_E(H)B and H X-permutes with B and with all subgroups (with all Sylow subgroups, respectively) V of B such that (|H|, |V |) = 1. We analyze the influence of X-quasipermutable and XS-quasipermutable subgroups on the structure of G. In particular, it is proved that if every Sylow subgroup P of G is F(G)-quasipermutable in its normal closure PG in G, then G is supersoluble.