The influence of weakly Z-permutable subgroups on the structure of finite groups (original) (raw)

2016

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. A subgroup H of G is said to be Z-permutable of G if H permutes with every member of Z. A subgroup H of G is said to be a weakly Z-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H ∩K ≤ HZ, where HZ is the subgroup of H generated by all those subgroups of H which are Zpermutable subgroups of G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of prime order p or of order 4 (if p = 2) is a weakly Z-permutable subgroup of G. Our results extend and generalize several results in the literature.

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