On strongly closed and Hall s-semiembedded subgroups of finite groups

Abstract

Let [Formula: see text] be subgroups of a finite group [Formula: see text], then [Formula: see text] is called strongly closed in [Formula: see text] with respect to [Formula: see text] if [Formula: see text] for every [Formula: see text], and in particular, [Formula: see text] is simply called strongly closed in [Formula: see text] if [Formula: see text] is strongly closed in [Formula: see text] with respect to [Formula: see text]. Let [Formula: see text] be a subgroup of a finite group [Formula: see text], then [Formula: see text] is called Hall [Formula: see text]-semiembedded in [Formula: see text] if [Formula: see text] is a Hall subgroup of [Formula: see text] for every [Formula: see text], where [Formula: see text]. In this paper, we obtain some criteria for [Formula: see text]-nilpotency of a finite group and extend some known results concerning strongly closed and Hall [Formula: see text]-semiembedded subgroups. In particular, we generalize some main results of Guo and Li [Hall [Formula: see text]-semiembedded subgroups and [Formula: see text]-nilpotency of finite groups, Southeast Asian Bull. Math. 42(3) (2018) 367–374] and Kong [New characterizations of [Formula: see text]-nilpotency of finite groups, J. Algebra Appl. 20(11) (2021) Article ID: 2150215, 6 pp.].