Related papers: Cluster automorphism groups and automorphism group…

For a large class of finite graphs $E$, we show that whenever $\alpha$ is a vertex-fixing quasi-free automorphism of the corresponding graph $C^*$-algebra $C^*(E)$ such that $\alpha({\mathcal D}_E) \neq{\mathcal D}_E$, where ${\mathcal…

Cycle prefix digraphs have been proposed as an efficient model of symmetric interconnection networks for parallel architecture. It has been discovered that the cycle prefix networks have many attractive communication properties. In this…

If $G$ is a free product of finite groups, let $\Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em…

By an automorphism of a topological group G we mean an isomorphism of G onto itself which is also a homeomorphism. In this article, we study the automorphism group Aut(G) of a dense subgroup G of R^n, n>=1. We show that Aut(G) can be…

Let $K$ be a field of characteristic $0$ and let $G$ and $H$ be connected commutative algebraic groups over $K$. Let $\text{Mor}_0(G,H)$ denote the set of morphisms of algebraic varieties $G \to H$ that map the neutral element to the…

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of…

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic,…

Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $\Gamma (G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on…

To each symmetrizable Cartan matrix, we associate a finite free EI category. We prove that the corresponding category algebra is isomorphic to the algebra defined in [C. Geiss, B. Leclerc, and J. Schr\"{o}er, Quivers with relations for…

Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and…

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $ (G,\mathcal{M}) $, such that $ G $ is a finite 2-group and $ \mathcal{M}…

Let $G = H\times A$ be a group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \geq 2$, we show that there exists an isomorphism $\phi : Aut_{Z(G)}^{\gamma_{n}(G)}(G)…

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the…

Let $A$ be an infinitely generated free abelian group. We prove that the automorphism group $\aut A$ first-order interprets the full second-order theory of the set $|A|$ with no structure. In particular, this implies that the automorphism…

We prove that $Aut({\mathbb S}^1)$ coincides with the automorphism group of the \emph{circle graph} $\mathcal{C}$, i.e. the intersection graph of the family of chords of ${\mathbb S}^1$. We prove that the countable subgraph of $\mathcal{C}$…

We study the (virtual) indicability of the automorphism group $Aut(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$ associated to a simplicial graph $\Gamma$. First, we identify two conditions -- denoted (B1) and (B2) -- on $\Gamma$…

We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ induces an isomorphism between the homology groups…

Let $Aut_{alg}(X)$ be the subgroup of the group of regular automorphisms $Aut(X)$ of an affine algebraic variety $X$ generated by all connected algebraic subgroups. We prove that if $dim X \ge 2$ and if $Aut_{alg}(X)$ is rich enough,…

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of…