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

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it…

In a finite real reflection group, the reflection length of each element is equal to the codimension of its fixed space, and the two coincident functions determine a partial order structure called the absolute order. In complex reflection…

A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper,…

Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were…

In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation…

We study finite groups $G$ such that the maximum length of an orbit of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is bounded from above by a constant. Our main results are the following: Firstly, a finite…

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with…

In this paper, we classify the groups of semisimilarities of finite classical polar spaces with exactly two orbits on the singular or isotropic points. As a byproduct, we obtain many highly symmetric regular sets in the point graphs of…

We consider a group $GL(\infty)$ and its parabolic subgroup $B$ corresponding to partition $\infty=\infty+m+\infty$. Denote by $P$ the kernel of the natural homomorphism $B\to GL(m)$. We show that the space of double cosets of $GL(\infty)$…

A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime p, we call a bi-Cayley graph over a metacyclic p-group a bi-p-metacirculant.…

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph $\bcay(G,S)$ is called…

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that…

We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley…

If $G$ is a group of permutations of a set $\Omega$, then the suborbits of $G$ are the orbits of point-stabilisers $G_\alpha$ acting on $\Omega$. The cardinalities of these suborbits are the subdegrees of $G$. Every infinite primitive…

Let $\Ga = (V, E)$ be a graph and $a, b$ nonnegative integers. An $(a, b)$-regular set in $\Ga$ is a nonempty proper subset $D$ of $V$ such that every vertex in $D$ has exactly $a$ neighbours in $D$ and every vertex in $V \setminus D$ has…

Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $\Gamma=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as small as possible} if $\mathrm{Aut}(\Gamma)=…

We prove a conjecture of Medvedev and Scanlon in the case of regular morphisms of semiabelian varieties. That is, if $G$ is a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$, and $\varphi\colon G\to…

Dual representations $V$ and $V^*$ of a complex connected algebraic group $G$ simultaneously have either infinitely or finitely many orbits. Whenever the latter holds, the orbits in $V$ and $V^*$ are in a bijective correspondence called…

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this…

Let $X$ be a homogeneous space of a real reductive Lie group $G$. It was proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(X)$ contains each irreducible representation of $G$ at most finitely many times if a…