Factorization of finite simple groups

Cyclic factorization numbers of finite groups

In this paper we introduce and study the concept of cyclic factorization number of a finite group G. By using the Mobius inversion formula and other methods involving the cyclic subgroup structure, this is explicitly computed for some important classes of finite groups.

Factorizations of almost simple orthogonal groups in odd dimension

This is the third one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups in odd dimension.

Factorizations of almost simple orthogonal groups of minus type

This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus type.

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter…

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.

几乎单群的精确分解

The Exact Factorizations of Almost Simple Groups

本文对几乎单群的精确分解进行了分类，解决了自1980年以来的长期未决问题，并将其应用于双交叉积Hopf代数和置换群。

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。

有限分解性质由下幺半群检测

Finite factorization is detected by undermonoids

本文证明了对于可消交换幺半群，所有子幺半群都是有限分解幺半群当且仅当所有下幺半群都是有限分解幺半群。

Factorization graphs of finite groups

We introduce the factorization graph of a finite group and study its connectedness and forbidden structures. We characterize all finite groups with connected factorization graphs and classify those with connected bipartite factorization graphs. Also, we obtain a classification of all groups with claw-free, $K_{1,4}$-free, and square-free factorization graphs, excluding the non-solvable groups in the latter case.

168阶与360阶单群的因子分解

Factorizations of simple groups of order 168 and 360

本文证明了168阶和360阶单群具有多重可因子分解性质，并进一步猜想任意对称群S_n及n≥6的交替群A_n也满足该性质。

关于有限群分解的一个答案

Some answers regarding factorizations of finite groups

针对《库罗夫卡笔记本》中关于有限群分解的问题，G. M. 伯格曼给出\(k = 3\)反例并提问，本文给出\(k \geq 3\)的反例，解决了部分问题，\(k = 2\)时问题仍待解决。

A note on factorizations of finite groups

In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group $G$ and a factorization $\mathrm{card}(G)= n_1\ldots n_k$, one can always find subsets $A_1,\ldots,A_k$ of $G$ with $\mathrm{card}(A_i)=n_i$ such that $G=A_1\ldots A_k;$ equivalently, such that the group multiplication map $A_1\times\ldots\times A_k\to G$ is a bijection. We show that for $G$ the alternating group on 4 elements, $k=3$, and $(n_1,n_2,n_3) = (2,3,2)$, the answer is negative. We then generalize some of the tools used in our proof, and note an open question.

有限群的因子分解

Factorizations of finite groups

研究有限群的k-因子可分解性质，证明了对于k≥3存在非k-因子可分解的有限群，以及在特定条件下有限群不存在特定形式的因子分解，同时指出最多8个阶不超过100的群不满足k-因子可分解。

有限群的均匀循环群分解

Uniform Cyclic Group Factorizations of Finite Groups

本文提出均匀群分解概念，证明有限可解群均有均匀循环群分解，并揭示该问题与有限单群均匀分解的等价性，附有实例。

3-辫子拟正因子分解的 Hurwitz 等价判据

Criterion of Hurwitz equivalence for quasipositive factorizations of 3-braids

针对 3-辫子的拟正因子分解，本文提出判定有限代表集中两个元素是否在同一 Hurwitz 轨道的算法，从而给出 Hurwitz 等价判据。

有限群的完全分解

Complete factorizations of finite groups

本文研究有限群的完全分解问题，证明了任意有限幂零群在阶可分解为至少三个大于1的整数因子时，都存在相应大小的完全分解子集，并给出构造实例与开放问题。

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…

有限群中的因子分解

Factorizations in finite groups

给出了有限群$G$的元素的因子分解唯一的一个必要条件，其中因子属于$G$的某些共轭类的并。这个条件是充分的，如果属于每个共轭类的因子的数量足够大。将所得结果应用于给定伽罗瓦群$G$和固定局部单值集的$\mathbb P^1$的$d$次标记覆盖的Hurwitz空间的不可约分支的个数问题.

无因子分解性质的超线性群的例子

Examples of hyperlinear groups without factorization property

本文给出了一个局部可嵌入有限群的群的例子（特别是它最初是subsub-affordable的，sofic的，因此是超线性的），但不具有Kirchberg因子分解性质.这个群体也提供了一个例子sofic Kazhdan群这是不是剩余有限的，回答了一个问题的Elek和萨博。我们还给出了一个例子，一个组，这是不是最初subsubsubsub顺从，但超线性。最后指出了Kirchberg的一个错误，并给出了一个不具有因子分解性质的群仍是有限维连通李群的子群的例子.

有限格上弱分解系统的格结构研究

On the lattice of the weak factorization systems on a finite lattice

该研究针对有限格上的弱分解系统格展开，证明其半分配、trim且同余一致，提出枚举转移系统的图论方法，并给出布尔格上转移系统数的下界。

证明：在有限实反射群中，对每个抛物型拟Coxeter元素，其反射分解的Hurwitz作用轨道由两个显然不变量区分

In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants

本文证明有限Coxeter群中抛物型拟Coxeter元素的反射分解的Hurwitz轨道由生成子群和共轭类多重集完全区分，并分类了满足极小生成集为最小大小的有限Coxeter群。

类型 $B$ 和 $D$ 中反射因子分解的 Hurwitz 等价性

Hurwitz Equivalence of Reflection Factorizations in Types $B$ and $D$

本文证明了 Lewis 关于有限 Coxeter 群中反射因子分解的 Hurwitz 等价性猜想在类型 A、B、D 下成立，并给出了将猜想约化为例外类型有限问题的约化方法。

1-Cayley图的因子分解

1-Factorizations of Cayley graphs

本文证明了有限群$Q \times H$的所有连通Cayley图都是1-可因子分解的，其中$Q$是任意2-幂阶非平凡群，$H$是任意奇阶群.

A note on non-reduced reflection factorizations of Coxeter elements

We extend a result of Lewis and Reiner from finite Coxeter groups to all Coxeter groups by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.

On tensor factorizations of Hopf algebras

We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automo…

Unitarizability, Maurey--Nikishin factorization, and Polish groups of finite type

Let $Γ$ be a countable discrete group, and let $π\colon Γ\to {\rm{GL}}(H)$ be a representation of $Γ$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H\rtimes_πΓ$ is SIN ($G$ admits a two-sided invariant metric compatible with its topology) and unitarily representable ($G$ embeds into the unitary group $\mathcal{U}(\ell^2(\mathbb N))$), if and only if $π$ is uniformly bounded, and that $π$ is unitarizable if and only if $G$ is of finite type: that is, $G$ embeds into the unitary group of a II$_1$-factor. Consequently, we show that a unitar…

Property (T), property (F) and residual finiteness for discrete quantum groups

We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg's theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness. We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (facto…