Permutability of subgroups of G×H that are direct products of subgroups of the direct factors

If M and S are two subgroups of a group G, M and Spermute if MS=SM. Furthermore, M is a permutable subgroup of G if M permutes with every subgroup of G. We give necessary and sufficient conditions for M, a subgroup of G, to permute with a subgroup of G×H given that G and H are finite groups. The main part of the paper involves the development of a characterization of permutable subgroups of G×H that are direct products of subgroups of the direct factors; that is, subgroups that are equal to A×B where A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \leqq $\end{document}G and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \leqq $\end{document}H.