Related papers: On groups with automorphisms whose fixed points ar…

We prove that if a linear group $G$ is almost Engel, then $G$ is finite-by-hypercentral. If $G$ is almost nil, then $G$ is finite-by-nilpotent.

Suppose that a finite group $G$ admits an automorphism $\varphi $ of order $2^n$ such that the fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the involution $\varphi ^{2^{n-1}}$ is nilpotent of class $c$. Let $m=|C_G(\varphi)|$ be the…

Let $G$ be a finite group admitting a coprime automorphism $\alpha$. Let $J_G(\alpha)$ denote the set of all commutators $[x,\alpha]$, where $x$ belongs to an $\alpha$-invariant Sylow subgroup of $G$. We show that $[G,\alpha]$ is soluble or…

Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question…

Given an affine algebraic variety $X$, we prove that if the neutral component $\mathrm{Aut}^\circ(X)$ of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves…

Let $A$ be a set and $f:A\rightarrow A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $*$ such that $(A,*)$ is a cyclic group and $f\in…

We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $\bar a$, $\bar b \in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$. We further…

We obtain some general restrictions on the continuous endomorphisms of a profinite group G under the assumption that G has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if G is…

Let $Ab_0$ be the class of finite abelian groups and consider the function $f:Ab_0\longrightarrow(0,\infty)$ given by $f(G)=\frac{|{\rm Aut}(G)|}{|G|}$\,, where ${\rm Aut}(G)$ is the automorphism group of a finite abelian group $G$. In this…

An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents…

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Suppose that $G$ is residually finite and the subgroup $[G,H] =…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant…

In this paper, we prove that the endomorphism rings End A and End A' of periodic infinite Abelian groups A and A' are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all…

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism…

By a coprime commutator in a profinite group $G$ we mean any element of the form $[x, y]$, where $x,y\in G$ and $(|x|,|y|)=1$. It is well-known that the subgroup generated by the coprime commutators of $G$ is precisely the pronilpotent…

In this note, we prove: \medskip \noindent {\bf Theorem A:} \emph{ There is a fixed constant $C$ such that for any positive integer $n$ and prime $p$, every finite subgroup $G$ of order coprime to $p$ of ${\rm GL}(n,\mathbb{C})$ has an…

The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a…

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually…

Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by…

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.