Commuting involutions and elementary abelian subgroups of simple groups

Finite groups with many involutions

It is shown that a finite group in which more than 3/4 of the elements are involutions must be an elementary abelian 2-group. A group in which exactly 3/4 of the elements are involutions is characterized as the direct product of the dihedral group of order 8 with an elementary abelian 2-group.

具有两个交换对合的扩展特殊线性群 $ES{L_3}(\mathbb{Z})$, $ES{L_2}(\mathbb{Z})$ 的对合极小生成集以及 GL$_2$($\mathbb{F}_p$), SL$_3(\mathbb{F}_p)$ 中的根公式 \RomanNumeralCaps{3}

Involutive minimal generating sets with two commuting involutions of Extended Special Linear group $ES{L_3}(\mathbb{Z})$, $ES{L_2}(\mathbb{Z})$ and analytical and recursive formulas of roots in SL$_3(\mathbb{F}_p)$ \RomanNumeralCaps{3}

研究扩展特殊线性群ESL3(Z)和ESL2(Z)中两个可交换对合生成的极小生成集，并给出GL2(Fp)和SL3(Fp)中根的解析与递推公式。

Explicit constructions of loops with commuting inner mappings

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product $xy$ with $xyh$ in certain positions, where $h$ is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.

交错群由对合生成的字符串群的最大秩

The maximal rank of a string group generated by involutions for alternating groups

本文研究了由对合生成的字符串群（SGGI）的秩，确定了$n$次交错群上SGGI秩的上界，并证明了当$n\equiv 0,1,4\pmod 5$时该上界是紧的。

Recognizing Right-Angled Coxeter Groups Using Involutions

We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. We provide some first applications. In addition, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover a resul…

A note on commuting automorphisms of some finite $p$-groups

An automorphism $α$ of a group $G$ is called a commuting automorphism if each element $x$ in $G$ commutes with its image $α(x)$ under $α$. Let $A(G)$ denote the set of all commuting automorphisms of $G$. Rai [Proc. Japan Acad., Ser. A {\bf 91} (2015), no. 5, 57-60] has given some sufficient conditions on a finite $p$-group $G$ such that $A(G)$ is a subgroup of Aut$(G)$ and, as a consequence, has proved that in a finite $p$-group $G$ of co-class 2, where $p$ is an odd prime, $A(G)$ is a subgroup of Aut$(G)$. We give here very elementary and short proofs of main results of Rai.

Structure of Hyperbolic Unitary Groups II: Classification of E-normal Subgroups

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group $U_{2n}(R,Λ)$ which are normalized by the elementary subgroup $EU_{2n}(R,Λ)$, under the condition that $R$ is a quasi-finite ring with involution, i.e a direct limit of module finite rings with involution, and $n\geq 3$.

Finite groups with very few character values

Finite groups with very few character values are characterized. The following is the main result of this article: a finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a non-trivial elementary abelian $3$-group. The proof involves the analysis of the centralizers of involutions.

Degree of commutativity of infinite groups

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.

奇素数阶元素的Brauer-Fowler界

Brauer-Fowler Bounds for Elements of Odd Prime Order

本文研究奇素数阶元素的Brauer-Fowler界，证明Skresanov的对合相关结果的直接类似物对奇素数阶元素不成立，给出了有限单群阶数由中心化子内p阶元素数量及指数界定的结论。

局部环上${}^2E_6$型扭Chevalley群的自同构

Automorphisms of twisted Chevalley groups of type ${}^2E_6$ over local rings

本文分类了带对合且2可逆的交换局部环上，两种等义类型的${}^2E_6$型扭Chevalley群的抽象自同构，证明其均为标准型，明确了伴随与单连通情形下自同构的结构，通过基变换替换等步骤完成分类。

关于具有可解中心化子的有限群

On finite groups with soluble centralisers

本文分类了特定非中心元素中心化子可解的有限群，给出全可解情形的结构描述、π-元素情形的归约定理，成果可应用于相关群类与非交换图问题。

Palindromes and orderings in Artin groups

The braid group $B_{n}$, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism ${\rm{rev}}: B_{n} \to B_{n}$, $v \mapsto \bar{v}$, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $τ:x \mapsto Δ^{-1}x Δ$ by the generalized half-twist (Garside element). More generally, the involution ${\rm{rev}}$ is defined for all Artin groups (equipped with Artin's presentation) and the involution $τ$ is defined for all Artin groups of finite type. A palindrome is an element invariant un…

对合生成群上的Jensen泛函方程：平方根准则、刚性现象与障碍空间

Jensen's Functional Equation on Involution-Generated Groups: A Square-Root Criterion, Rigidity Phenomena, and Obstruction Spaces

本文研究对合生成群上的Jensen泛函方程，引入平方根准则$(\mathrm{SR}_2)$，揭示相关刚性现象，构建障碍空间理论，证明实正交群$O(n)$满足$(\mathrm{SR}_2)$，明确归一化解的同态性质。

The Multiple Holomorphs of Finite $p$-Groups of Class Two

$\DeclareMathOperator{\Hol}{Hol}$$\DeclareMathOperator{\Aut}{Aut}$$\newcommand{\Gp}[0]{\mathcal{G}(p)}$$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$Let $G$ be a group, and $S(G)$ be the group of permutations on the set $G$. The (abstract) holomorph of $G$ is the natural semidirect product $\Aut(G) G$. We will write $\Hol(G)$ for the normalizer of the image in $S(G)$ of the right regular representation of $G$, \begin{equation*} \Hol(G) = N_{S (G)}(ρ(G)) = \Aut(G) ρ(G) \cong \Aut(G) G, \end{equation*} and also refer to it as the holomorph of $G$. More generally, if $N$ is any regular subg…

Small loops of nilpotency class three with commutative inner mapping groups

Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Csörgő type. In order to obtain small loops of Csörgő type, we expand our programme from `Explicit constructions of loops with commuting inner mappings', European J. Combin. 29 (2008), 1662-1681, and analyze the following setup in groups: Let $G$ be a group, $Z\le Z(G)$, and suppose that $δ:G/Z\times G/Z\to Z$ satisfies $δ(x,x)=1$, $δ(x,y)=δ(y,x)^{-1}$, $z^{yx}δ([z,y],x) = z^{xy}δ([z,x],y)$ for every $x$, $y$, $z\…

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。

Frobenius-Schur Indicators for Subgroups and the Drinfel'd Double of Weyl Groups

If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgr…

Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group W is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied in Boothby et al.. In particular the authors enumerated cyclically fully commutative elements in all Coxeter groups having a finite number of them. In this work we characterize and enumerate cyclically fully commutative elements according to their Coxeter length in all finite or aff…

关于李型群的可约 Killing 型

On reducible Killing forms for groups of Lie type

本文研究 Lie 秩一有限单李型群中对合或幂幺元共轭类所对应 Killing 型的非退化性与不可约性，并揭示其与特征标理论和交换图的联系。

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups $U_{2n+1}(R,Δ)$. These groups contain as special cases the odd dimensional general linear groups $GL_{2n+1}(R)$ where $R$ is any ring, the odd dimensional orthogonal and symplectic groups $O_{2n+1}(R)$ and $Sp_{2n+1}(R)$ where $R$ is any commutative ring and further the first author's even dimensional unitary groups $U_{2n}(R,Λ)$ where $(R,Λ)$ is any form ring. We classify the E-normal subgroups of the groups $U_{2n+1}(R,Δ)$ (i.e. the subgroups which are normalized by the elementary subgroup $EU_{2n+1}(R,Δ)$), under the condition that $R$ …