Finite groups with n-embedded subgroups

Let [Formula: see text] be a finite group. How minimal subgroups can be embedded in [Formula: see text] is a question of particular interest in studying the structure of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for all Sylow subgroups [Formula: see text] of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] with [Formula: see text]-embedded subgroups.

FINITE GROUPS WITH COMPLEMENTED 2-MINIMAL -SUBGROUPS

For a given prime $p$, we investigate the finite groups all of whose 2-minimal $p$-subgroups are complemented.

Finite groups with some minimal subgroups are ℋC-subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text], if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], for all [Formula: see text]. In this paper, we investigate the structure of groups based on the assumption that every subgroup of [Formula: see text] of order [Formula: see text] or 4 (if [Formula: see text]) is an [Formula: see text]-subgroup of [Formula: see text], here [Formula: see text] is a Sylow [Formula: see text]-subgroup of [Formula: see text]. Some results for a group to be [Formula: see text]-nilpotent and supersolvable are obtained and many known results are generalized.