Related papers: On commuting and non-commuting complexes

We study finite groups $G$ with elements $g$ such that $\lvert \mathbf{C}_G(g)\rvert = \lvert G:G' \rvert$. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class…

Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

In this paper we define and investigate a class of groups characterized by a representation-theoretic property we call purely noncommuting or PNC. This property guarantees that the group has an action on a smooth projective variety with…

The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod…

The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property…

A base of a permutation group (X,G) is a subset B of X such that its pointwise stabilizer is the trivial group. A list (x1,x2, ... ,xk) of elements of X is irredundant if each element is not in the pointwise stabilizer of its predecessors.…

In this paper, we investigate certain generalizations of Camina pairs. Let $H$ be a nontrivial proper subgroup of a finite group $G$. We first show that every nontrivial irreducible complex character of $H$ induces homogeneously to $G$ if…

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy…

A subset X of a group G is a set of pairwise non-commuting ele- ments if ab 6= ba for any two distinct elements a and b in X. If jXj ? jY j for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset…

Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $\Xi(G)$ of $G$ has vertex set $G \setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y] \ne 1$ and $\langle x, y…

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the…

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph $\Gamma_G$ associated to $G$ is the graph whose vertex set is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only if $xy\neq yx$.…

We show that for some classes of groups $G$, the homotopy fiber $E_{\mathrm{com}} G$ of the inclusion of the classifying space for commutativity $E_{\mathrm{com}} G$ into the classifying space $BG$, is contractible if and only if $G$ is…

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double…

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$…

In this paper, we investigate the question of how one can recover the homology of a simplicial complex $X$ equipped with a regular action of a finite group $G$ from the structure of its quotient space $X/G.$ Specifically, we describe a…

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such…

In this paper we describe a classifying theory for families of simplicial topological groups. If $B$ is a topological space and $G$ is a simplicial topological group, then we can consider the non-abelian cohomology $H(B,G)$ of $B$ with…

A non-commuting graph of a finite group $G$ is a graph whose vertices are non-central elements of $G$ and two vertices are adjacent if they don't commute in $G$. In this paper, we study the non-commuting graph of the group $U_{6n}$ and…