Related papers: Cayley groups

A connected linear algebraic group G is called a Cayley group if the Lie algebra of G endowed with the adjoint G-action and the group variety of G endowed with the conjugation G-action are birationally G-isomorphic. In particular, the…

A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give conditions for when a Cayley graph on an abelian group can be represented as a…

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e. a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra Lie(G). A prototypical example is the classical…

We show that for certain integers $n$, the problem of whether or not a Cayley digraph $\Gamma$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a…

We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.

A linear algebraic group G is over a field K is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We prove that the special unitary group in 3…

New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…

In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where…

For any positive integer $n$ we describe the Leavitt path algebra of the Cayley graph $C_n$ corresponding to the cyclic group $\Z/n\Z$. Using a Kirchberg-Phillips-type realization result, we show that there are exactly four isomorphism…

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and the Lie algebra Lie(G). A Cayley map can be thought of as a…

For an integer $n\geq 2$, let $X_n$ be the Cayley graph on the symmetric group $S_n$ generated by the set of transpositions ${(1 2),(1 3),...,(1 n)}$. It is shown that the spectrum of $X_n$ contains all integers from $-(n-1)$ to $n-1$…

A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define $\mathcal{M}$ to be a \emph{Cayley extension} of…

Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical…

Given positive integers $k$ and $n$, we present methods to construct all groups of order at most $n$ that contain a Cayley set of size $k$, and to enumerate the Cayley sets of order $k$ in a given group, up to the action of the automorphism…

The classical Cayley transform is a birational map between a quadratic matrix group and its Lie algebra, which was first discovered by Cayley in 1846. Because of its essential role in both pure and applied mathematics, the classical Cayley…

A linear algebraic group G is over a field K is called a Cayley K-group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups…

Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can…

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are…

A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the…

Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$.…