Related papers: Counting periodic points over finite fields

Let $\sigma$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $\sigma$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points…

For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$…

It is proved that when R is a local ring of positive characteristic, $\phi$ is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through…

We classify by numerical invariants the finite subgroups $H$ of a primary abelian group $G$ for which every homomorphism or monomorphism of $H$ into $G$, or every endomorphism of $H$, extends to an endomorphism of $G$. We apply these…

Let $S$ be a class of groups and let $f_S (n)$ be the number of isomorphism classes of groups in $S$ of order $n$. Let $f(n)$ count the number of groups of order $n$ up to isomorphism. The asymptotic bounds for $f(n)$ behave differently…

We continue some recent investigations of W. Dziobiak, J. Jezek, and M. Maroti. Let G=(G,\cdot) be a commutative group. A semilattice over G is a semilattice enriched with G as a set of unary operations acting as semilattice automorphisms.…

Suppose $F$ is either a global field or a finitely generated extension of ${\mathbf Q}$, $A$ is an abelian variety over $F$, and $\ell$ is a prime not equal to the characteristic of $F$. Let $Z$ denote the center of the endomorphism algebra…

We prove that if f is a self-map of an algebraic variety over a field K, then under certain conditions on X, f and K the set of possible periods of K-valued periodic points of f is finite.

Let $V$ be a $G$-module where $G$ is a complex reductive group. Let $Z:=\quot VG$ denote the categorical quotient and let $\pi\colon V\to Z$ be the morphism dual to the inclusion $\O(V)^G\subset\O(V)$. Let $\phi\colon Z\to Z$ be an…

We adapt the theory of partition rank and analytic rank to the category of abelian groups. If $A_1, \dots, A_k$ are finite abelian groups and $\phi : A_1 \times \cdots \times A_k \to \mathbf{T}$ is a multilinear map, where $\mathbf{T} =…

When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.

Let $G$ and $H$ be two simple graphs. A bijection $\phi:V(G)\rightarrow V(H)$ is called an isomorphism between $G$ and $H$ if $(\phi v_i)(\phi v_j)\in E(H)$ $\Leftrightarrow$ $v_i v_j\in E(G)$, $\forall v_i,v_j \in V(G)$. In the case that…

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures…

Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property…

In this paper, we investigate structural properties of finite groups that are detected by certain group invariants arising from Dijkgraaf--Witten theory, a topological quantum field theory, in one space and one time dimension. In this…

Let G be a group of permutations of a denumerable set E. The profile of G is the function phi which counts, for each n, the number phi(n) of orbits of G acting on the n-subsets of E. Counting functions arising this way, and their associated…

For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting…

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

Let \phi be an endomorphism of a finitely generated free group F, and let H be a finite-index subgroup of F that is invariant under \phi. The nonzero eigenvalues of \phi are contained in the eigenvalues of \phi restricted to H.

We examine the different ways of writing a cover of curves $\phi\colon C\to D$ over a field $K$ as a composition $\phi=\phi_n\circ\phi_{n-1}\circ\dots\circ\phi_1$, where each $\phi_i$ is a cover of curves over $K$ of degree at least $2$…