Mutually Permutable Products of Finite Groups. (20th July 2011) - Hijazi, Rola A.
Abstract : Let G be a finite group and G 1, G 2 are two subgroups of G . We say that G 1 and G 2 are mutually permutable if G 1 is permutable with every subgroup of G 2 and G 2 is permutable with every subgroup of G 1 . We prove that ifG = G 1 G 2 = G 1 G 3 = G 2 G 3 is the product of three supersolvable subgroups G 1, G 2, and G 3, where G i and G j are mutually permutable for all i and j withi ≠ 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 subgroupsG i   ( i = 1, 2, 3 ) whose indices are pairwise relatively prime, and G i and G j are mutually permutable for all i and j withi ≠ j, then G is supersolvable.