Mutually Permutable Products of Finite Groups

Abstract EN

Let G be a finite group and G1, G2 are two subgroups of G.

We say that G1 and G2 are mutually permutable if G1 is permutable with every subgroup of G2 and G2 is permutable with every subgroup of G1.

We prove that if G=G1G2=G1G3=G2G3 is the product of three supersolvable subgroups G1, G2, and G3, where Gi and Gj are mutually permutable for all i and j with i≠j and the Sylow subgroups of G are abelian, then G is supersolvable.

As a corollary of this result, we also prove that if G possesses three supersolvable subgroups Gi (i=1,2,3) whose indices are pairwise relatively prime, and Gi and Gj are mutually permutable for all i and j with i≠j, then G is supersolvable.