相关论文: A finitely presented group with unbounded dead-end…

The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the…

The function $\mathrm{P}_{\mathbf{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes (as proved in [12]) only values $\frac{m-1}{m}$, for $1 \leq m \leq 6$, in the…

We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…

Let $G$ be a finite group and $f:G \to {\mathbb C}$ be a function. For a non-empty finite subset $Y\subset G$, let $I_Y(f)$ denote the average of $f$ over $Y$. Then, $I_G(f)$ is the average of $f$ over $G$. Using the decomposition of $f$…

Let A denote either the automorphism group of the free group of rank n>=4 or the mapping class group of an orientable surface of genus n>=12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the…

Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an…

For an element $g$ of a group $G$, a right Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[g ,x],x],\dots ,x]$ for all $x\in G$. A left Engel sink of $g$ is a subset of $G$ containing all…

T.C. Burness and S.D. Scott \cite{3} classified finite groups $G$ such that the number of prime order subgroups of $G$ is greater than $|G|/2-1$. In this note, we study finite groups $G$ whose subgroup graph contains a vertex of degree…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of G^n with respect to any generating set is at most n(|G|-rank(G)); and the second one states that there…

Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly…

Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and…

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where…

Let $N_{g,n}$ denote the nonorientable surface of genus $g$ with $n$ boundary components and $M(N_{g,n})$ its mapping class group. We obtain an explicit finite presentation of $M(N_{g,n})$ for $n=0,1$ and all $g$ such that $g+n>3$.

Groups of finite type (also called finitely constrained groups), introduced by Grigorchuk, are known to be the closure of regular branch groups. This article explores many of their properties. Firstly, we prove that being finitely…

Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is…

When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.

A group has finite palindromic width if there exists $n$ such that every element can be expressed as a product of $n$ or fewer palindromic words. We show that if $G$ has finite palindromic width with respect to some generating set, then so…

We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that `strongly converge' to the regular representation of the group. The corresponding statement…

The conjugator length function of a finitely generated group $\Gamma$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $\Gamma$. We…