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...

(1) Let G be a group and H ⊂ Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p2 (p a prime number) is abelian and that the centre of any non-abelian group of order p3 has p elements. (2) (?) Let H be a normal subgroup of a p-group G, H 6= {e}. Prove that H∩Z(G) 6= {e}. In particular, one obtains that any normal subgroup with p-elements is con...

We call a Cayley digraph Γ = Cay(G,S) normal for G if R(G), the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on the groups of order pq and also on non-abelian finite groups G such that every proper subgroup of G is abelian.

‎motivated by a celebrated theorem of schur‎, ‎we show that if~$gamma$ is a normal subgroup of the full automorphism group $aut(g)$ of a group $g$ such that $inn(g)$ is contained in $gamma$ and $aut(g)/gamma$ has no uncountable abelian subgroups of prime exponent‎, ‎then $[g,gamma]$ is finite‎, ‎provided that the subgroup consisting of all elements of $g$ fixed by $gamma$ has finite index‎. ‎so...