The Influence of C- Z-permutable Subgroups on the Structure of Finite Groups

Abstract

Let Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, Z contains exactly one and only one Sylow p-subgroup of G, say Gp. Let C be a nonempty subset of G. A subgroup H of G is said to be C-Z-permutable (conjugateZ-permutable) subgroup of G if there exists some x ∈ C such that HxGp = GpHx, for all Gp ∈ Z. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-Z-permutable subgroups of G.