Marius Tarnauceanu
This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the Möbius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving the results of {\it Factorization numbers of some finite groups}, Glasgow Math. J. (2012).
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This is the first one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple linear groups.
This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor.
In this short note we give a formula for the factorization number $F_2(G)$ of a finite rank $3$ abelian $p$-group $G$. This extends a result in our previous work \cite{9}.
This is the second one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple unitary 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.
This is the fifth 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 plus type.
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.
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.
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.
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…
We prove that all Mathieu groups, some linear, and unitary groups are factorizable.
We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group $G$ has such a factorization if and only if $G$ is a central product of $\left\langle X\right\rangle $ and $\left\langle Y\right\rangle $ and the central subgroup $\left\langle X\right\rangle \cap \left\langle Y\right\rangle $ sat…
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。
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。
Hecke Algebra-valued Poincaré Series and Geometric Factorization of Affine Weyl Groups
本文通过测地线管及其双曲稳定子的几何研究仿射外尔群，引入关联框架证明相关猜想，为单连通群建立Hecke代数值ζ函数恒等式的部分结果，凸显三者间相互作用。
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。
Near-factorizations of dihedral groups
本文研究二面体群等非交换群的近似分解，证明已知构造多产生等价分解，并借助计算机发现新示例，同时分析Pêcher方法以探究获得不等价分解的条件。
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.
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。
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。
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.
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。
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。
Let $G$ be a finite group and let $A_1,\ldots,A_k$ be a collection of subsets of $G$ such that $G=A_1\ldots A_k$ is the product of all the $A_i$'s with $|G|=|A_1|\ldots|A_k|$. We write $G=A_1\cdot\ldots\cdot A_k$ and call this a $k$-fold factorization of $G$ of the form $(|A_1|,\ldots,|A_k|)$ or more briefly an $(|A_1|,\ldots,|A_k|)$-factorization of $G$. Let $k\geq2$ be a fixed integer. If $G$ has an $(a_1,\ldots,a_k)$-factorization, whenever $|G|=a_1\ldots a_k$ with $a_i>1$, $i=1,\ldots,k$, we say that $G$ is $k$-factorizable. We say that $G$ is multifold-factorizable if $G$ is $k$-factoriza…
We consider the subgroup lpG_{k,1} of length preserving elements of the Thompson-Higman group G_{k,1} and we show that all elements of G_{k,1} have a unique lpG_{k,1}.F_{k,1} factorization. This applies to the Thompson-Higman group T_{k,1} as well. We show that lpG_{k,1} is a ``diagonal'' direct limit of finite symmetric groups, and that lpT_{k,1} is a k^infinity Pr"ufer group. We find an infinite generating set of lpG_{k,1} which is related to reversible boolean circuits. We further investigate connections between the Thompson-Higman groups, circuits, and complexity. We show that elements of …
It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups without nontrivial elements of order $\le n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.
Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to triangular coordinates is polynomial. We show that the inverse is rational. There is an algorithm for the inverse (involving LDU factorization), and a related explicit formula for Haar measure in root subgroup coordinates. In classical cases there are preferred reduced factorizations of the longest Weyl group elements, and conjecturally in these cases there …
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。
Let $F$ be any finite-rank free group, and $R$ be any finite subset of $\{g, [g]: g \in F-\{1\}\}$, where $[g]:= \{fgf^{-1}:f\in F\}$. By an $R$-allocating $F$-factorization we mean a set $\mathcal{H}$ of nontrivial subgroups of $F$ such that $\ast_{H \in \mathcal{H}} H = F$ and $R \subseteq \{h, [h] : h \in H, H\in \mathcal{H}\}$. We show that Whitehead's (fast) cutvertex algorithm inputs the pair $(F,R)$ and outputs a maximum-size $R$-allocating $F$-factorization. Richard Stong showed this in the case where $R \subseteq F$ or $R \subseteq \{[g] : g \in F\}$, thereby unifying and generalizing…
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。
We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d, 1, n)$ (the group of $d$-colored permutations) and $G(d, d, n)$.
$λ$-fold near-factorizations of groups
本文首次系统研究$λ>1$的群的$λ$-重近因子分解，建立基本性质与等价概念，证明必要条件，给出无限族构造并刻画子族，同时提供小阶阿贝尔群的计算结果。
In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question of Elek and Szabo. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in an assertion of Kirchberg and provide an example of a group which does not have the factorization property and is still a subgro…
In [11] we showed that a loop in a simply connected compact Lie group $\dot{U}$ has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group $\dot G_0$ of inner type. In [4] we showed that for an element of $\dot G_0$, i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a r…
Inspired by Ol'shanskii's work, we provide an axiomatic framework to describe certain irreducible unitary representations of non-discrete unimodular totally disconnected locally compact groups. We then look at the applications to certain groups of automorphisms of locally finite trees and semi-regular right-angled buildings.
Cyclotomic construction of $λ$-fold near-factorizations of cyclic groups
本文利用分圆方法，在素数 $p=4n^4+12n^2+1$ 对应的循环群 $\mathbb{F}_p$ 中构造了每个非单位元素恰好出现 $\lambda$ 次的 $\lambda$-重近因子分解。
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.
The aim of the paper is to give a full classification of factorizations of groups in terms of descent cohomology (pointed) sets introduced in [5]. We show that descent cohomology includes Serre's non-abelian group cohomology as a special case. This enables us to generalize Serre's theory further to include monoids.
In this note we prove that all connected Cayley graphs of every finite group $Q \times H$ are 1-factorizable, where $Q$ is any non-trivial group of 2-power order and $H$ is any group of odd order.
We study the number of ways of factoring elements in the complex reflection groups G(r,s,n) as products of reflections. We prove a result that compares factorization numbers in G(r,s,n) to those in the symmetric group on n letters, and we use this comparison, along with the ELSV formula, to deduce a polynomial structure for factorizations in G(r,s,n).
Hurwitz Equivalence of Braid Group Factorizations Consisting of a Semi-Frame
本文证明了辫群的某些Hurwitz等价性质。我们的主要结果是：当分解的元素是半框架时，$Δ_n ^2$的每两个分解都是Hurwitz等价的.本文的结果是\cite{B4}中结果的推广。我们使用一种新的表示的辫群，称为Birman-Ko-Lee表示，定义半框架结构。本文的主要结果可用于计算曲面的BMT不变量。BMT是辫群的中心元的Hurwitz等价分解类。BMT区分不相互变形的同构曲面。
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。
We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain linear order of the generators that is compatible with the chosen element.
We study class $\mathcal S$ for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class $\mathcal S$ and prove unique prime factorization results for group von Neumann algebras of products of locally compact groups in this class. We also prove that class $\mathcal S$ is a measure equivalence invariant.
We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis and the question whether there is an anlogue of the well known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis and Reiner as well a…
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.
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.
Uniqueness and explicit computation of mates in near-factorizations
本文证明了有限群近因子分解中配对集的唯一性，并给出显式高效计算公式，据此排除阶小于200的非循环阿贝尔群中非平凡近因子分解的存在，同时找到若干指标为2的例子。
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…
Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems.
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。
Triangular and Unitriangular Factorization of Twisted Chevalley Groups
本文针对扭曲Chevalley群中未解决的${}^2A_{2n}$型情形，引入特殊稳定秩一条件和$\theta$-完备环两类新环类，证明了在该类环上三角与单位三角分解的存在性，涵盖所有域、局部环等。
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。
We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in $G(m, p, n)$.
Graph Theoretic Method for Determining non Hurwitz Equivalence in the Braid Group and Symmetric group
受辫群中$Δ^2$分解的Hurwitz等价问题的启发，我们讨论了对称群中的Hurwitz等价问题，该问题是通过将$Δ^2$分解投影到$S_n$中得到的.我们得到了带有转置因子的$1_{S_n}$因子分解。将转置看作图中的边，我们证明了两个因子分解是Hurwitz等价的当且仅当它们的图具有相同的加权连通分量。本文的主要结果将有助于我们计算“辫单值型”不变量。图结构给出了一个较弱但很容易计算的不变量，以区分彼此不变形的同构曲面。
Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have…
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…
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