Related papers: The generating graph of a profinite group

We describe a flexible construction that produces triples of finitely generated, residually finite groups $M\hookrightarrow P \hookrightarrow \Gamma$, where the maps induce isomorphisms of profinite completions…

A group $G$ is said to satisfy the finitely generated intersection property (f.g.i.p.) if the intersection of any two finitely generated subgroups of $G$ is again finitely generated. The aim of this article is to understand when the…

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we…

We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\equiv_{\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating…

Let $G$ be a finite group and $\text{cd}(G)$ denote the character degree set for $G$. The prime graph $\Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $\text{cd}(G)$, denoted $\rho(G)$. Two primes…

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if $\Gamma$ is a finite graph which contains at least two vertices and is not a join and…

Let R be any ring (with 1), \Gamma a group and R\Gamma the corresponding group ring. Let H be a subgroup of \Gamma of finite index. Let M be an R\Gamma -module, whose restriction to RH is projective. Moore's conjecture: Assume for every…

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in…

A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random…

A group $\Gamma$ has separable cohomology if the profinite completion map $\iota \colon \Gamma \to \widehat{\Gamma}$ induces an isomorphism on cohomology with finite coefficient modules. In this article, cohomological separability is…

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key…

Let $n$ be a positive integer, $q$ be a prime power, and $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$. Let $G := V \rtimes G_0$, where $G_0$ is an irreducible subgroup of ${\rm GL}(V)$ which is maximal by inclusion with…

A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take…

For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there…

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…

We prove that every finitely generated residually finite group $G$ can be embedded in a finitely generated branch group $\Gamma$ such that two elements in $G$ are conjugate in $G$ if and only if they are conjugate in $\Gamma$. As an…

Consider a graph $\Gamma$. 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 containing cycles. If $\Gamma$ has a cyclic vertex cutset, then it…

The codegree of an irreducible character $\chi$ of a finite group $G$ is defined as $|G:\ker\chi|/\chi(1)$. The codegree graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, where two distinct…

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 containing cycles. If $\Gamma$ has a cyclic…