Related papers: Regular and semi-regular representations of groups…

A recent conjecture of the author and Teng Fang states that there are only finitely many finite simple groups with no cubic graphical regular representation. In this paper, we make a crucial progress towards this conjecture by giving an…

Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. We prove that if $G$ is a soluble group with finite…

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for…

Let $G$ be a permutation group acting on a finite set $\Omega$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $\Omega_m$ of all size $m$ subsets of $\Omega$ satisfies the trivial inequalities…

Let $G$ be a group acting on a set $X$ of combinatorial objects, with finite orbits, and consider a statistic $\xi : X \to \mathbb{C}$. Propp and Roby defined the triple $(X, G, \xi)$ to be \emph{homomesic} if for any orbits $\mathcal{O}_1,…

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of $\textrm{Aut}(G)$. We study…

We show that every finite simple group is generated invariably by a Sylow subgroup and a cyclic group. It follows that that the order complex of the coset poset of an arbitrary finite group has nontrivial reduced rational homology.

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such…

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one…

Given any representation V of a complex linear reductive Lie group G_0, we show that a larger semi-simple Lie group G with g=g_0 + V + V* + ..., exists precisely when V has a finite number of G_0-orbits. In particular, V admits an open…

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do…

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on…

Let G be a real or complex linear algebraic reductive group. Let H and F be reductive subgroups. We study the natural H action on G/F. The main theorem of this note shows that generic H orbits are closed. This theorem is then applied to…

Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems…

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it…

We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be…

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group…

Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism types of a sporadic simple group G, the outer automorphism group Out(G) has order 2, and in the…

Distance-regular graphs are a class of regualr graphs with pretty combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the problem of charaterizing distance-regular Cayley graphs, which can be viewed as a natural…