On σ-semipermutable Subgroups of Finite Groups

Abstract

Let σ = {σ i |i ∈ I} be some partition of the set of all primes P, G a finite group and σ(G) = {σ i |σ i ∩ π(G) = ∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member = 1 of H is a Hall σ i -subgroup of G for some σ i ∈ σ and H contains exact one Hall σ i -subgroup of G for every σ i ∈ σ(G).

Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups

“…It is well known that permutable subgroups and supplemented subgroups play an important role in the theory of finite groups. Recall that a subgroup A of G is said to be σ-semipermutable in G if G possesses a complete Hall σ-set H such that AH x = H x A for all x ∈ G and all H ∈ H with σ(A) ∩ σ(H) = ∅ (see [19]). Let τ H (A) = {σ i ∈ σ(G)\σ(A) | σ(A) ∩ σ(H G ) ̸ = ∅ for a Hall σ i -subgroup H ∈ H}.…”

“…A subgroup A of G is said to be weakly σ-permutable in G if G has a σ-subnormal subgroup T such that G = AT and A ∩ T ≤ A σG , where A σG is the subgroup of A generated by all those subgroups of A which are σ-permutable in G (see [42]). By using the above subgroups and supplemented subgroups, the researchers have obtained a series of interesting results (see, for example, [4,6,8,10,14,19,26,27,31,34,38,42]). Now, we consider the following new generalized supplemented subgroup.…”