The p-nilpotency of finite groups with some CSS-subgroups.

A subgroup H of a finite group G is said to be C S S -subgroup of G if there exists a normal subgroup K of G such that G = H K and H ∩ K is S S -quasinormal in G. In this paper, we investigate the p -nilpotency of the finite groups under the assumption that there is a subgroup D of P such that 1 < | D | < | P | and every subgroup of P with order | D | or 2 | D | (whenever | D | = 2) is C S S -subgroup, where p is a prime dividing the order of G and P is a Sylow p -subgroup of G. As an application of the results, a related problem posed by Heliel et al. in [Finite groups with minimal CSS-subgroups, Eur. J. Pure Appl. Math.14 (2021) 1002–1014] is solved. [ABSTRACT FROM AUTHOR]