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ISRN Algebra
Volume 2011 (2011), Article ID 867082, 4 pages
http://dx.doi.org/10.5402/2011/867082
Research Article

Mutually Permutable Products of Finite Groups

Rola A. Hijazi

Department of Mathematics, Faculty of Science 14466, King Abdulaziz University, Jeddah 21424, Saudi Arabia

Received 19 June 2011; Accepted 10 July 2011

Academic Editors: M. Asaad, G. Buskes, G. Mason, and L. Vinet

Copyright © 2011 Rola A. Hijazi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 = 𝐺 1 𝐺 2 = 𝐺 1 𝐺 3 = 𝐺 2 𝐺 3 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 β‰  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 𝐺 𝑖 ( 𝑖 = 1 , 2 , 3 ) whose indices are pairwise relatively prime, and Gi and Gj are mutually permutable for all i and j with 𝑖 β‰  𝑗 , then G is supersolvable.