Pronormal subgroups of a direct product of groups

Abstract

We give criteria to characterize abnormal, pronormal and locally pronormal subgroups of a direct product of two finite groups A × B, under hypotheses of solvability for at least one of the factors, either A or B.

Embedding Properties in Central Products

“…One open question is to characterize pronormal subgroups of central products. Results on pronormal subgroups of direct products, as in [2], may be especially useful and inspirational for this direction of work.…”

“…In [2], abnormal subgroups were characterized for direct products of two groups, where one of the direct factors is solvable. This result is presented in the following lemma.…”

“…Recently, more work has been done on subgroups of a direct product of two groups. For example, in [3], [4], [1], and [2], necessary and sufficient conditions have been provided to characterize subgroups of a direct product including normal and subnormal subgroups. However, very little work has been done to characterize these subgroups for a central product of two groups.…”

The $p$-nilpotency of finite groups with some weakly pronormal subgroups

“…A subgroup H of a group G is called pronormal in G if for each g ∈ G, the subgroups H and H g are conjugate in H, H g . The pronormality is one of the most significant properties pertaining to subgroups of groups and has been studied widely, for example, see [1], [3], [5], [6], [9], [19], [20], [21]. Recently, Asaad introduced the following concept: Definition 1.1 ([1], Definition 1.1).…”