The influence of SS-quasinormality of some subgroups on the structure of finite groups

Abstract

The following concept is introduced: a subgroup H of the group G is said to be SS-quasinormal (Supplement-Sylow-quasinormal) in G if H possesses a supplement B such that H permutes with every Sylow subgroup of B. Groups with certain SS-quasinormal subgroups of prime power order are studied. For example, fix a prime divisor p of |G| and a Sylow p-subgroup P of G, let d be the smallest generator number of P and M d (P ) denote a family of maximal subgroups P 1 , . . . , P d of P satisfying d i=1 (P i ) = Φ(P ), the Frattini subgroup of P . Assume that the group G is p-solvable and every member of some fixed M d (P ) is SS-quasinormal in G, then G is p-supersolvable.

On solvability of finite groups with some ss-supplemented subgroups

“…Recall that a subgroup H of a group G is said to be s-permutable in G if H permutes with every Sylow subgroup P of G, that is, HP = P H (see [13]); H is said to be c-supplemented in G if G has a subgroup K such that G = HK and H ∩ K H G , where H G is the normal core of H in G (see [3]); H is said to be ss-quasinormal in G if there is a subgroup K of G such that G = HK and H permutes with every Sylow subgroup of K (see [14]). Recently, Guo and Lu in [7] introduced the following concept, which covers both the ss-quasinormality and c-supplementation concepts.…”

Finite groups with ℋ-subgroups or strongly closed subgroups

“…In [26,33], Li, Shen and Shi considered a subset M d ðPÞ of MðPÞ for a given Sylow p-subgroup P of G defined in the following way: Definition 1.5. Let d be the smallest number of generators of a p-group P and M d ðPÞ ¼ fP 1 ; .…”

Finite groups with normally embedded subgroups

“…In [14], Li and Shen considered another generalization of S-quasinormal subgroups and gave the following definition: Definition 1.1 (see [14]). Let G be a group.…”

“…In [14], Li and Shen considered a subset M d ðPÞ of MðPÞ for a given Sylow p-subgroup P of G, defined in the following way: Definition 1.2 (see [14]). Let d be the smallest number of generators of a p-group P and let M d ðPÞ ¼ fP 1 ; .…”

“…In [14], Li and Shen showed that if for every prime p dividing the order of G and P A Syl p ðGÞ, every member of some fixed M d ðPÞ is SS-quasinormal in G, then G is supersolvable. However, there exist a saturated formation F containing U and a solvable group G with a normal p-subgroup P such that G=P A F and such that every member of some fixed M d ðPÞ is S-quasinormal (and hence SS-quasinormal) in G, but with G B F.…”