In this paper, in addition to some elementary facts about the ultra-groups, which their structure based on the properties of the transversal of a subgroup of a group, we focus on the relation between a group and an ultra-group. It is verified that every group is an ultra-group, but the converse is not true generally. We present the conditions under which, for every normal subultra-group of an u...

Abstract A subgroup X of a group G is called transitively normal if in any Y such that $$X\le Y$$ X≤Y and subnormal . Thus all subgroups are only normality transitive relation every (i.e. $$\overline{T}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:...

Let G be a finite group with an abelian normal subgroup N . When does N have a unique conjugacy class of complements in G? We consider this question with a focus on subgroups and properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup (Theorem 1 of [3]). Furthermore, w...