Bilinear maps and central extensions of abelian groups

有限域上双线性映射的自同构群的平均阶

On average orders of automorphism groups of bilinear maps over finite fields

本文研究有限域上双线性映射在一般线性群作用下的同构类数量渐近界，并将其应用于 Frattini 类 2 的 p-群计数、平凡自同构群矩阵空间采样及立方零交换代数同构类估计。

Alternate modules are subsymplectic

In this paper, an alternate module $(A,ϕ)$ is a finite abelian group $A$ with a $\mathbb{Z}$-bilinear application $ϕ:A\times A\rightarrow \mathbb{Q}/\mathbb{Z}$ which is alternate (i.e. zero on the diagonal). We shall prove that any alternate module is subsymplectic, i.e. if $(A,ϕ)$ has a Lagrangian of cardinal $n$ then there exists an abelian group $B$ of order $n$ such that $(A,ϕ)$ is a submodule of the standard symplectic module $B\times B^*$.

Abelian covers of graphs and maps between outer automorphism groups of free groups

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with a…

Elementary abelian regular coverings of Platonic maps, Case I: ordinary representations

We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the representations of G on certain homology groups of the sphere, punctured at the branch-points. We give a complete classification for branching over faces (or, dually, vertices) of M, and outline how the method extends to other branching patterns.

带有双线性型的向量空间中可定义集合、群与域

Sets, groups, and fields definable in vector spaces with a bilinear form

本文研究带双线性型的向量空间理论中可定义群与域的结构，引入维数概念，证明群具有（代数-由-阿贝尔）-由-代数结构，域同构于标量域，并推广到实闭域情形。

Virtually abelian quotients of random groups

It is well-known that random groups with at least as many relators as generators have vanishing first betti number with high probability. In this paper we extend this question and study maps from few-relators random groups to infinite virtually abelian groups.

Distortion for Abelian Subgroups of $\mathrm{Out}(F_n)$

We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for $\mathrm{Out}(F_n)$.

通过交换映射构造斜 bracoid 及 Yang-Baxter 方程的解

Constructing skew bracoids via abelian maps, and solutions to the {Y}ang-{B}axter equation

本文提出利用交换映射构造的斜 brace 生成斜 bracoid 族，并在特定条件下为 Yang-Baxter 方程提供右非退化解。

String numbers of abelian groups

The string number of self-maps arose in the context of algebraic entropy and it can be viewed as a kind of combinatorial entropy function. Later on its values for endomorphisms of abelian groups were calculated in full generality. We study its global version for abelian groups, providing several examples involving also Hopfian abelian groups. Moreover, we characterize the class of all abelian groups with string number zero in many cases and discuss its stability properties.

Protori 与无挠阿贝尔群

Protori and Torsion-Free Abelian Groups

本文利用紧致阿贝尔群分解定理证明 protorus 的 profinite 子群构成格，引入非阿基米德维数作为分类不变量，并研究其与无挠阿贝尔群的联系。

Abelian maps, bi-skew braces, and opposite pairs of {H}opf-{G}alois structures

Let $G$ be a finite nonabelian group, and let $ψ:G\to G$ be a homomorphism with abelian image. We show how $ψ$ gives rise to two Hopf-Galois structures on a Galois extension $L/K$ with Galois group (isomorphic to) $G$; one of these structures generalizes the construction given by a ``fixed point free abelian endomorphism'' introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equat…

Abelian maps, brace blocks, and solutions to the {Y}ang-{B}axter equation

Let $G$ be a finite nonabelian group. We show how an endomorphism of $G$ with abelian image gives rise to a family of binary operations $\{\circ_n: n\in \mathbb Z^{\ge 0}\}$ on $G$ such that $(G,\circ_m,\circ_n)$ is a skew left brace for all $m,n\ge 0$. A brace block gives rise to a number of non-degenerate set-theoretic solutions to the Yang-Baxter equation. We give examples showing that the number of solutions obtained can be arbitrarily large.

Chirality Groups of Maps and Hypermaps

Although the phenomenon of chirality appears in many investigations of maps and hypermaps no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants -- the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are calle…