几乎单群的精确分解

几乎单群的分解及其应用

Factorizations of Almost Simple Groups with Applications

本文解决有限几乎单群分解分类中遗留的主要情形，确定所有满足H和K恰有一个非可合成分因子的有限几乎单典型群分解，完成该分类并得出双完美双交叉积Hopf代数最小维数为41287680的结论。

Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups

A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the cycles, every non-bipartite connected 3-arc-transitive Cayley graph of a solvable group is a cover of the Petersen graph or the Hoffman-Singleton graph.

On solvable factors of almost simple groups

Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Li and Xia describe the factorizations of almost simple groups with a solvable factor $H$. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of $G_0$ containing $H \cap G_0$ is iden…

有限群的精确分解数量研究

On the number of exact factorization of finite Groups

该文研究有限群精确分解计数函数$f_2(G)$，计算了若干有限群族的$f_2(G)$，推导了交错群$A_{2^n}$精确分解数的渐近式，并提出后续研究问题。

几乎单群的可解因子的完全分类

A complete classification of solvable factors of almost simple groups

本文完成了几乎单群因子分解中可解因子的分类，解决了精确因子分解问题的最后剩余情形，并由此给出一维传递群的新刻画及拟本原置换群中可解传递子群“小”的结论。

A factorization result for classical and similitude groups

For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representations to its dual.

群的$(λ,μ)$-重分解理论

The theory of $(λ,μ)$-fold factorizations of groups

受相关研究启发，本文引入有限群的$(λ,μ)$-重分解，建立其基础理论、给出两种通用构造并证明有限阿贝尔群的若干不存在性结果。

Almost quasi-isometries and more non-exact groups

We study permanence results for almost quasi-isometries, the maps arising from the Gromov construction of finitely generated random groups that contain expanders (and hence that are not C*-exact). We show that the image of a sequence of finite graphs of large girth and controlled vertex degrees under an almost quasi-isometry does not have Guoliang Yu's property A. We use this result to broaden the application of Gromov's techniques to sequences of graphs which are not necessarily expanders. Thus, we obtain more examples of finitely generated non-C*-exact groups.