Related papers: A Generalization of Redfield's Master Theorem

Let $\Gamma =(V,E)$ be a reflexive relation having a transitive group of automorphisms and let $v\in V.$ Let $F$ be a subset of $V$ with $F\cap \Gamma ^-(v)=\{v\}$. (i) If $F$ is finite, then $| \Gamma (F)\setminus F|\ge |\Gamma (v)|-1.$…

This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss…

For a classical group $G$ of type $\mathsf D_n$ over a field $k$ of characteristic different from $2$, we show the existence of a finitely generated regular extension $R$ of $k$ such that $G$ admits outer automorphisms over $R$. Using this…

Take a bounded symmetric domain $D$ and an arithmetic subgroup $\Gamma$ of ${\rm Aut}(D)$. Take the quotient $D/\Gamma$, compactify and resolve the singularities. We study the fundamental group of the compact complex manifolds that result…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

Let $k$, $\lambda$ and $\mu$ be positive integers. A decomposition of a multigraph $ \lambda G$ into edge-disjoint subgraphs $G_1, \ldots , G_k$ is said to be \emph{enclosed} by a decomposition of a multigraph $\mu H$ into edge-disjoint…

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further…

This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and…

In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.

The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in…

We give a technical result that implies a straightforward necessary and sufficient conditions for a graph of groups with virtually cyclic edge groups to be one ended. For arbitrary graphs of groups, we show that if their fundamental group…

Let $G$ be a finite group, let $x \in G$, and let $p$ be a prime. We prove that the commutator $[x,g]$ is a $p$-element for every $g \in G$ if and only if $x$ is central modulo $\mathbf{O}_p(G)$, where $\mathbf{O}_p(G)$ denotes the largest…

For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^…

A fine mixed subdivision of a (d-1)-simplex T of size n gives rise to a system of ${d \choose 2}$ permutations of [n] on the edges of T, and to a collection of n unit (d-1)-simplices inside T. Which systems of permutations and which…

The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a…

This is the first in a sequence of papers that will develop the theory of automorphisms of nonsolvable finite groups. The sequence will culminate in a new proof of McBride's Nonsolvable Signalizer Functor Theorem, which is one of the…

We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that…

Let $(\Omega, \leq)$ be a totally ordered set. We prove that if $\Aut(\Omega,\leq)$ is transitive and satisfies the same first-order sentences as $\Aut(\RR,\leq)$ (in the language of lattice-ordered groups) then $\Omega$ and $\RR$ are…

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared…

We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case…