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# On Mutually $m$-permutable Product of Smooth Groups

Published:2014-03-03
Printed: Jun 2014
• A. M. Elkholy,
Beni Suef University, Faculty of Science, Mathematics Department, Beni-Suef 62511, Egypt
• M. H. Abd El-Latif,
Beni Suef University, Faculty of Science, Mathematics Department, Beni-Suef 62511, Egypt
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## Abstract

Let $G$ be a finite group and $H$, $K$ two subgroups of G. A group $G$ is said to be a mutually m-permutable product of $H$ and $K$ if $G=HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group which is a mutually m-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.
 Keywords: permutable subgroups, $m$-permutable, smooth groups, subgroup lattices
 MSC Classifications: 20D10 - Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] 20D20 - Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 20E15 - Chains and lattices of subgroups, subnormal subgroups [See also 20F22] 20F16 - Solvable groups, supersolvable groups [See also 20D10]