A result of Dixon, Evans and Smith shows that if G is a locally (soluble-by- finite) group whose proper subgroups are (finite rank) -by-abelian, then G itself has this property, i.e. the Commutator subgroup of G has finite rank. It is proved here that if G is a locally (soluble-by- finite) group whose proper subgroups have minimax Commutator subgroup, then also the Commutator subgroup G’ of G is minimax. A corresponding result is proved for groups in which the Commutator subgroup of every proper subgroup has finite torsion-free rank.

We introduce a new regular relation δ on a given group G and show that δ is a congurence relation on G, with respect to module the Commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

We introduce the notions of commuting automorphism and Commutator polygroups. The basic question that can be arose about the set of all commuting automorphisms is that for the assumed polygroup (P, ·), under what conditions the set of all commuting automorphism A(P) is a subgroup of Aut(P). In this paper basically the answer to this question is investigated for the class of polygroups.

The neutrosophic automorphisms of a neutrosophic groups  G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of  G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the sequence G (I), X1, X2, … is the identity, then G (I) is nilpotent.

We call G to be an α,-coset group, if it contains a proper α,-invariant normal subgroup N such that Nxα,= {xg | g ∈,G}, for some automorphism α,of G and any x ∈,G\N. Clearly, if α,is identity automorphism of G, one obtains the notion of con-cos groups, which was first introduced by Muktibodh in 2006. In the present article, we discuss some properties of the new notion. Also, we introduce the concept of α,-Camina groups and give its connection with the groups of property P, where P is the class of all finite groups such that their α,-centres are the same as α,-Commutator subgroups of order p.

Let $w=w(x_1,\ldots,x_r)$ be an outer Commutator word‎. ‎We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-Commutator words in disjoint sets of variables‎. ‎This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$‎, ‎where the $n_i$ are non-zero integers‎. ‎Our approach is via the study of values of $w$ on normal subgroups‎, ‎and in this setting, ‎we obtain the following result‎: ‎if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all‎ ‎values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite, ‎then also the subgroup generated by these values‎, ‎i.e. $w(N_1,\ldots,N_r)$‎, ‎is finite‎.