A note on factorizations of finite groups

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…

Factorization of finite simple groups

We prove that all Mathieu groups, some linear, and unitary groups are factorizable.

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.

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。

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.

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。

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。

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.

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。

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。

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。

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.

Factorizations in finite groups

A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each conjugacy class is big enough. The result is applied to the problem on the number of irreducible components of the Hurwitz space of degree $d$ marked coverings of $\mathbb P^1$ with given Galois group $G$ and fixed collection of local monodromies.

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。

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。

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…