On τσ-quasinormal subgroups of finite groups

Abstract

Let D ¹ i j i 2 Iº be a partition of the set of all primes P and G a finite group. A set H of subgroups of G is said to be a complete Hall -set of G if every member ¤ 1 of H is a Hall i-subgroup of G for some i 2 I and H contains exactly one Hall i-subgroup of G for every i such that i \ .G/ ¤ ;. Let H .A/ D ¹ i 2 .G/n .A/ j .A/\ .H G/ ¤ ; fora Hall i-subgroup H of Gº. We say that a subgroup A of G is -permutable or -quasinormal in G with respect to H if AH x D H xA for all x 2 G and all H 2 H such that .H/ H .A/, and -permutable or -quasinormal in G if A is -permutable in G with respect to some complete Hall -set of G. We study G assuming that -quasinormality