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 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.

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.

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space over the field $\mathbb{C}$, $\mathcal{B}(\mathcal{H})$ be the $\ast$-algebra of all linear bounded operators on $\mathcal{H}$, let $|X|=\sqrt{X^*X}$ for $X\in \mathcal{B}(\mathcal{H})$. An operator $A\in \mathcal{B}(\mathcal{H})$ is a COMMUTATOR if $A=[S, T]=ST-TS$ for some $S, T\in \mathcal{B}(\mathcal{H})$. Let $X, Y \in \mathcal{B}(\mathcal{H})$ and $X\geq 0$. If the operator $XY$ is a non-COMMUTATOR, then $X^pYX^{1-p}$ is a non-COMMUTATOR for every $00$ then $|A|^q$ is a non-COMMUTATOR for every $q>0$. Let $\mathcal{H}$ be separable and $A \in \mathcal{B}(\mathcal{H})$ be a non-COMMUTATOR. If $A$ is hyponormal (or cohyponormal) then $A$ is normal. We also present results in the case of a finite-dimensional Hilbert space.

Purpose: The purpose of this paper is to study the multilinear COMMUTATOR in weighted Morrey space.Methods: We divide the multilinear COMMUTATOR into four parts and apply with the methods of weighted Lebesgue space and Perez and Trujillo-Gonzalez.Results: Let b®= (b1, …, bm), bj ∈ BMO (Rn), 1£j£ m; we show that T[1]b is bounded on weighted Morrey spaces Lp,κ (w), 1<p<¥, 0<k<1, w∈Ap. When p=1, it is shown that T®b is weak type bounded on weighted Morrey space.Conclusions: We extend the results of the COMMUTATOR in weighted Morrey space. In particular, for m=1, this result is also new in the classical COMMUTATOR.

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.