A product decomposition for the classical quasisimple groups

Constructive membership testing in black-box classical groups

The research described in this note aims at solving the constructive membership problem for the class of quasisimple classical groups. Our algorithms are developed in the black-box group model; that is, they do not require specific characteristics of the representations in which the input groups are given. The elements of a black-box group are represented, not necessarily uniquely, as bit strings of uniform length. We assume the existence of oracles to compute the product of two elements, the inverse of an element, and to test if two strings represent the same element. Solving the constructive…

Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups

Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as for the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete de…

群嵌入圈积：笛卡尔分解方法

Group Embedding into Wreath Products: Cartesian Decomposition Approach

本文利用笛卡尔分解，将正则群和对角群嵌入到乘积作用下的圈积中，提出了一种新的嵌入方法。

On free decompositions of verbally closed subgroups of free products of finite groups

For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) to be verbally closed.

Product decompositions in finite simple groups

We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of Lie type of bounded rank. Some of our arguments apply recent advances in the theory of growth in finite simple groups of Lie type, and provide a variety of new product decompositions of these groups.

坐标识别：一般理论、群及其他惊喜

Coordinate recognition: General theory, Groups, and other surprises

本文提出结构类识别坐标的性质，给出其等价刻画，并证明该性质在约化积上导致同构提升和强量词消去等结果，同时考察了群等具体结构类。

Finding direct product decompositions in polynomial time

A polynomial-time algorithm is produced which, given generators for a group of permutations on a finite set, returns a direct product decomposition of the group into directly indecomposable subgroups. The process uses bilinear maps and commutative rings to characterize direct products of p-groups of class 2 and reduces general groups to p-groups using group varieties. The methods apply to quotients of permutation groups and operator groups as well.

Disjoint direct product decomposition of permutation groups

Let $H \leq S_n$ be an intransitive group with orbits $Ω_1, Ω_2, \ldots ,Ω_k$. Then certainly $H$ is a subdirect product of the direct product of its projections on each orbit, $H|_{Ω_1} \times H|_{Ω_2} \times \ldots \times H|_{Ω_k}$. Here we provide a polynomial time algorithm for computing the finest partition $P$ of the $H$-orbits such that $H = \prod_{c \in P} H|_c$ and demonstrate its usefulness in some applications.

Rigidity of graph products of abelian groups

We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.

Quasiprimitive groups and blow-up decompositions

The blow-up construction by L. G. Kovács has been a very useful tool to study embeddings of finite primitive permutation groups into wreath products in product action. In the present paper we extend the concept of a blow-up to finite quasiprimitive permutation groups, and use it to study embeddings of finite quasiprimitive groups into wreath products.

On the product decomposition conjecture for finite simple groups

We prove that if $G$ is a finite simple group of Lie type and $S$ a subset of $G$ of size at least two then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of Lie type of bounded rank.

Intransitive Cartesian decompositions preserved by innately transitive permutation groups

We study Cartesian decompositions of sets that are acted upon intransitively by innately transitive permutation groups. We prove that such groups have at most three orbits on such a decomposition. A consequence of this result is that if $G$ is an innately transitive subgroup of a wreath product in product action then the natural projection of $G$ into the top group has at most 2 orbits.

Semidirect product decomposition of Coxeter groups

Let $(W,S)$ be a Coxeter system, let $S=I \dot{\cup} J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\widetilde{W}$ be the subgroup of $W$ generated by $\widetilde{J}$. We show that $W=\widetilde{W} \rtimes W_I$ and that $(\widetilde{W},\widetilde{J})$ is a Coxeter system.

Decompositions de groupes par produit direct et groupes de Coxeter

We provide examples of groups which are indecomposable by direct product, and more generally which are uniquely decomposable in direct products of indecomposable groups. Examples include Coxeter groups, for which we give an alternative approach to recent results of L. Paris. For a finitely generated linear group Gamma, we establish an upper bound on the number of factors of which Gamma can be the direct product. If Gamma is moreover torsion-free, it follows that Gamma is uniquely decomposable in direct product of indecomposable groups.

Linear decomposition attack on public key exchange protocols using semidirect products of (semi)groups

We show that a linear decomposition attack based on the decomposition method introduced by the author works by finding the exchanged secret keys in all main protocols using semidirect products of (semi)grops proposed by Kahrobaei, Shpilrain, Habeeb, Koupparis and Lam.

Tensor product decompositions of II$_1$ factors arising from extensions of amalgamated free product groups

In this paper we introduce a new family of icc groups $Γ$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II$_1$ factor $L(Γ)$ arise only from the canonical direct product decompositions of the underlying group $Γ$. Our groups are assembled from certain HNN-extensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As …

Decomposition of tensor products of modular irreducibles for $\SL_2$

We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed.

Unique decomposition of tensor products of irreducible representations of simple algebraic groups

We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.

Generators for Decompositions of Tensor Products of Modules associated with standard Jordan partitions

If $K$ is a field of finite characteristic $p$, $G$ is a cyclic group of order $q=p^α$, $U$ and $W$ are indecomposable $KG$-modules with $\dim U=m$ and $\dim W=n$, and $λ(m,n,p)$ is a standard Jordan partition of $ m n$, we describe how to find a generator for each of the indecomposable components of the $KG$-module $U \otimes W$.

Product decompositions of quasirandom groups and a Jordan type theorem

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-…

完全图笛卡尔积的传递路径分解

Transitive path decompositions of Cartesian products of complete graphs

本文研究完全图笛卡尔积 $K_n \Box K_n$ 的传递路径分解，当 $n$ 为奇素数时构造了路径长度相对于顶点数较大的分解，支持了 Gallai 猜想和 Ringel 猜想的扩展版本。

关于Saxl图的公共邻点猜想在每个基大小下均不成立

Common neighbour conjectures for Saxl graphs fail at every base size

本文证明了Saxl图的公共邻点猜想在所有基大小下均不成立，构造了对应反例并否定回答了《Kourovka笔记》的相关问题，同时完成了该猜想在特定仿射群下的成立性证明。

Decomposition of Pauli groups via weak central products

For any $m \ge 1$ and odd prime power $\mathtt{q}=\mathtt{p}^m$, for $\mathtt{q}=2$, and for any $n \ge 1$, we show a result of decomposition for Pauli groups $\mathcal{P}_{n,\mathtt{q}}$ in terms of weak central products. This can be used to describe the underlying structure of Pauli groups on $n$ qudits of dimension $\mathtt{q}$ and enables us to identify abelian subgroups of $\mathcal{P}_{n,\mathtt{q}}$. As a consequence of our main results, we show a similar factorisation for the so--called `lifted' Pauli groups, recently introduced by Gottesman and Kuperberg in the context of error-correc…

Decomposition of tensor products of modular irreducible representations for $SL_3$ (With an Appendix by C.M. Ringel)

We give an algorithm for working out the indecomposable direct summands in a Krull--Schmidt decomposition of a tensor product of two simple modules for G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family. Along the way we obtain the submodule structure of various Weyl and tilting modules. Some of the tilting modules that turn up in characteristic 3 are not rigid; these seem to provide the first example of non-rigid tilting modules for algebraic gr…

Decomposition theorems for asymptotic property C and property A

We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition complexity. Moreover, finite APC-decomposition complexity implies property A for metric spaces. We also show that finite APC-decomposition complexity is preserved by direct products of groups and spaces, amalgamated products of groups, and group extensions, among other constructions.

SU(1,1) 的全纯离散级数：正交关系、特征标公式与张量积分解中的重数

Holomorphic Discrete Series of SU(1,1): Orthogonality Relations, Character Formulas, and Multiplicities in Tensor Product Decompositions

本文重新审视 SU(1,1) 的全纯离散级数表示，重点研究矩阵元的正交关系、幺正不可约表示的特征标公式及张量积分解中的重数，深化了对 SU(1,1) 对称框架的数学基础与物理意义的理解。