X-semipermutable subgroups of finite groups

Abstract

Let X be a non-empty subset of a group G. Then we call a subgroup A of G a X-semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T 1 of T there exists an element x ∈ X such that AT x 1 = T x 1 A. In this paper, we study the properties of X-semipermutable subgroups. In particular, a new version of the famous Schur-Zassenhaus Theorem in terms of X-semipermutable subgroups is given.

On Existence of Hall Subgroups in Finite Groups

“…Let A, B be subgroups of a group G and X a non-empty subset of G. Following [6,7], we say that A is X-permutable with B if there exists some x ∈ X such that AB x = B x A. Our main results are the following: Theorem A.…”

τ-primitive subgroups of finite groups

“…Let A and B be subgroups of a group G and let X be a nonempty subset of G. Following [10], we say that A is X-permutable with B if there exists some x ∈ X such that AB x = B x A. The following lemma is evident.…”

On nonnilpotent groups in which every two 3-maximal subgroups are permutable

“…For instance, Guo and Shum proved [10] the solubility of groups all whose 2-maximal subgroups have the cover and avoidance property. In [11,12] supersoluble groups are characterized in terms of 2-maximal subgroups. Observe also that [13] gives a description of the nonnilpotent groups in which every 2-maximal subgroup permutes with every 3-maximal subgroup, as well as of the groups in which every maximal subgroup permutes with all 3-maximal subgroups [14].…”