Related papers: Classification of finite group automorphisms with …

Let $G$ be a group. A function $G\rightarrow G$ of the form $x\mapsto x^{\alpha}g$ for a fixed automorphism $\alpha$ of $G$ and a fixed $g\in G$ is called an affine map of $G$. In this paper, we study finite groups $G$ with an affine map of…

The goal of this paper is to generalize, refine, and improve results on large intersections. We show that if $G$ is a countable abelian group and $\varphi, \psi : G \to G$ are homomorphisms such that at least two of the three subgroups…

We study finite groups $G$ such that the maximum length of an orbit of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is bounded from above by a constant. Our main results are the following: Firstly, a finite…

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex.…

Let $G$ be a permutation group on a set $\Omega$ of size $t$. We say that $\Lambda\subseteq\Omega$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda$. We define the…

In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some…

Given a set of primes $\pi$, the $\pi$-index of an element $x$ of a finite group $G$ is the $\pi$-part of the index of the centralizer of $x$ in $G$. If $\pi=\{p\}$ is a singleton, we just say the $p$-index. If the $\pi$-index of $x$ is…

The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group isospectral to $L$ is…

Let $F$ be a free group of positive, finite rank and let $\Phi\in Aut(F)$ be a polynomial-growth automorphism. Then $F\rtimes_\Phi\mathbb Z$ is strongly thick of order $\eta$, where $\eta$ is the rate of polynomial growth of $\phi$. This…

A finite group $G$ is called *uniformly generated*, if whenever there is a (strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle x_1,x_2\rangle <\cdots<\langle x_1,x_2,\dots,x_d\rangle=G$, then $d$ is the minimal number of…

Let $\Gamma$ be an undirected and simple graph. A set $ S $ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is disconnected and has at least two components each containing a cycle. If $\Gamma$ has a…

A classical theorem states that the group of automorphisms of a manifold $M$ preserving a $G$-structure of finite type is a Lie group. We generalize this statement to the category of $cs$ manifolds and give some examples, some of which…

A graph $\Gamma$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\Gamma$ with $V(\Gamma)$ admitting…

A finite permutation group $G$ on $\Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $\Omega \times \Omega$. The largest permutation group on $\Omega$ having the same orbits as $G$ on $\Omega \times…

Let $\sigma=\{\sigma_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma(G)=\{\sigma_{i}|\sigma_{i}\cap \pi(|G|)\neq\emptyset\}$. A subgroup $S$ of a group $G$ is called a $\sigma_i$-sylowizer…

We use Sigma-invariants to study homotopical and homological finiteness properties of fixed subgroups of automorphisms of a group $G$ in terms of its center $Z(G)$ and the induced automorphisms on its associated quotient $G/Z(G)$.…

We describe the autotopism group Atp(G) of any abelian group G as being a semidirect product of its automorphism group Aut(G) and G^2. We then provide the subgroup structure of Atp(G) when G is a finite cyclic group.

This work provides an effective algorithm for distinguishing finite quotients between two non-isomorphic finitely generated Fuchsian groups $\Gamma$ and $\Lambda$. It will suffice to take a finite quotient which is abelian, dihedral, a…

Let $\Gamma$ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for $\Gamma$, determined on its elements of finite…

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by…