Permutable subgroups of a direct product

On characterizing permutability in direct products

“…The author has examined permutability in direct products in a series of papers [3], [4], and [5]. This paper concludes these investigations.…”

“…In [4], we investigate subgroups of G × H whose intersections with the direct factors are normal. It is in [5] that we put these results together to provide necessary and sufficient conditions for a subgroup of a direct product of finite groups to be permutable. Theorem 1.1 (Theorem 4.2 [5]).…”

“…It is in [5] that we put these results together to provide necessary and sufficient conditions for a subgroup of a direct product of finite groups to be permutable. Theorem 1.1 (Theorem 4.2 [5]). Let M be a subgroup of the finite pgroup G × H. Without loss of generality, assume exp…”

“…Despite the presence of this theorem, interesting questions remain. Specifically we present in Conjecture 5.1 of [5] a much more natural condition that is sufficient for a subgroup of a direct product to be permutable. Unfortunately, Example 5.4 in [5] demonstrates that this condition, presented here as Condition A, is not in general necessary for permutability of a subgroup M of a direct product…”

Pronormal subgroups of a direct product of groups

“…Particular details are included in Section 2. The normal, subnormal, permutable, CAP, system permutable and normally embedded subgroups of a direct product have been studied in several articles [3,[8][9][10][11]13,15]. For a survey article discussing various contributions to this research see [4].…”