On partially conjugate-permutable subgroups of finite groups

Abstract

Let R be a subset of a group G. We call a subgroup H of G the R-conjugate-permutable subgroup of G, if HHx = HxH for all x ∈ R. This concept is a generalization of conjugatepermutable subgroups introduced by T. Foguel. Our work focuses on the influence of R-conjugatepermutable subgroups on the structure of finite groups in case when R is the Fitting subgroup or its generalizations F ∗(G) (introduced by H. Bender in 1970) and F ˜(G) (introduced by P. Shmid 1972). We obtain a new criteria for nilpotency and supersoulbility of finite groups which generalize some well known results.