Finite groups with given systems of weakly S-propermutable subgroups

Abstract

Abstract A subgroup H of a finite group G is said to be S-propermutable in G provided there is a subgroup B of G such that G = N G ⁢ ( H ) ⁢ B ${G=N_{G}(H)B}$ and H permutes with all Sylow subgroups of B (see [18]). In this paper, we study the following generalization of the concept of S-permutability: A subgroup H of a finite group G is called weakly S-propermutable in G provided there is a K- 𝔘 ${\mathfrak{U}}$ -subnormal subgroup (see [1, p. 236]) T of G such that H ⁢ T = G ${HT=G}$ and H ∩ T ≤ S ≤ H ${H\cap T\leq S\leq H}$ for some S-propermutable subgroup S of G. Our main goal here is to prove the following result (Theorem 1.2): Let X be a normal subgroup of G and P a Sylow p-subgroup of X, where ( p - 1 , | X | ) = 1 ${(p-1,|X|)=1}$ . Suppose that there is a subgroup 1 < D < P ${1<D<P}$ such that every subgroup of P of order | D | ${|D|}$ and also, in the case when P is non-abelian and | D | = 2 ${|D|=2}$ , every cyclic subgroup of order 4 of P is weakly S-propermutable in G. Then X is p-nilpotent and every p-chief factor of G below X is cyclic. This theorem generalizes many known results and also gives the positive answer to Question 18.91 (b) in Kourovka Notebook [14].