PERMUTABLE DIAGONAL-TYPE SUBGROUPS OF $G\times H$

Abstract

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.

On characterizing permutability in direct products

“…First observe that y, m 1 , M = G. Thus, y ∩M = 1, since M is Core-free by (3) in Example 2.4. Hence, it follows from (4) in Example 2.4 that | y /( y ∩M )|=p 4 .…”

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

“…Suppose that M is a subgroup of G and (g, Example 2.4. Suppose that p is an odd prime, and let g, x i for i ∈ N such that 1 ≤ i ≤ p be elements of S p 4 , where g = (1, 2, 3, . .…”

“…In [3], the permutability of a subgroup of G × H that is a direct product of subgroups of the direct factors is explored. 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.…”

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].…”