On self-normalizing subgroups of finite groups

Abstract

Abstract. The aim of this paper is to characterize the classes of groups in which every subnormal subgroup is normal, permutable, or S-permutable in terms of the embedding of subgroups (or subgroups of prime-power order) in their normal, permutable, or S-permutable closure.

Finite groups with some NR-subgroups or $${\mathcal{H}}$$ -subgroups

“…We give new characterizations of soluble T -, PT -, PST-groups and minimal non-PST-groups in terms of NR-subgroups or H-subgroups. We will show the differences between these characterizations and the ones given in [3][4][5][6].…”

“…A subgroup of G is called s-permutable in G if it permutes with all Sylow subgroups of G. A group G is said to be a PST-group if s-permutability is a transitive relation in G. By a result of Kegel ( [4], Theorem 1.2.14(3)) PST-groups are exactly those groups where all subnormal subgroups are s-permutable. In the literature there are several characterizations of finite soluble T -groups, PT -groups and PST-groups (see [3][4][5][6]8]). …”

“…Since a non-abelian simple group is a T -group the first assertion is not true for the class of all T -groups (PT -and PST-groups). For some other characterizations of these groups see [3][4][5][6]. The structure of minimal non-T -groups, minimal non-PTgroups and minimal non-PST-groups was established by Robinson [16,17].…”

Finite groups with NR-subgroups or their generalizations

“…There are several characterizations in the literature of finite solvable T -groups, PT-groups and PST-groups (see [5]). In [6,26] Ballester-Bolinches, Esteban-Romero and Li characterized the classes of finite solvable T -groups, PT-groups or PST-groups in terms of embedding of subgroups (or subgroups of prime-power order) in their normal, permutable, or s-permutable closure.…”

“…Definition 1.2. Let G be a group and let H 6 G. The permutable closure A G .H / of H in G is the intersection of all permutable subgroups of G containing H (see [6]). A triple .G; H; K/ is said to be permutable special in G if K E H 6 G and A G .K/ \ H D K. A subgroup H is said to be a PR-subgroup of G (Permutable Restriction) if, whenever K is normal in H , the triple .G; H; K/ is permutable special in G.…”