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Abstract. Two subgroups M and S of a group G are said to permute, or M permutes with S, if MS = SM. Furthermore, M is a permutable subgroup of G if M permutes with every subgroup of G. In this article, we provide necessary and sufficient conditions for a subgroup of G × H, whose intersections with the direct factors are normal, to be a permutable subgroup.2000 Mathematics Subject Classification. 20D40.

If M and S are two subgroups of a group G, M and S permute 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 Ϲ G and B Ϲ H.

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This paper extends the author's earlier results regarding permutable subgroups of direct products. More specifically, a prior article characterizes when a subgroup of a direct product of finite groups is permutable, and this article improves that characterization.