Finite Groups with G-Permutable Schmidt Subgroups

Abstract

AbstractGiven a non-empty set X of a group G, a subgroup A is said to be X-permutable (respectively, X-h-permutable) with a subgroup B, if $$AB^x = B^xA$$ A B x = B x A , for some element $$x\in X$$ x ∈ X (respectively, $$x \in X \cap \langle A,B\rangle $$ x ∈ X ∩ ⟨ A , B ⟩ ). A subgroup A of a group G is said to be X-permutable (respectively, X-h-permutable) in G if A is X-permutable (respectively, X-h-permutable) with every subgroup of G. In this paper, we study the structure of a finite group G with all its Schmidt subgroups G-permutable (respectively, G-h-permutable).