On the subgroup permutability degree of the simple Suzuki groups

Products of Sylow subgroups in Suzuki and Ree groups

An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characterictic.

弱次正规子群与Baer-Suzuki定理的变体

Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

该研究针对有限群中的弱次正规$p$-子群展开分类工作，进一步刻画含循环弱次正规$p$-子群的有限群，并借助群元素阶的性质推导Baer-Suzuki定理的多种变体，丰富有限群论相关结论。

从Thompson到Baer-Suzuki：可解radical的一个尖锐刻画

From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical

我们证明了素数阶元$g$属于有限群（或更一般地说，线性群）的可解根$R(G)$当且仅当对每个$x\in G$，由$g, xgx^{-1}$生成的子群是可解的.这个定理意味着有限（或线性）群$G$是可解的当且仅当在$G$的每个共轭类中每两个元素生成一个可解子群。

The Möbius Function of the Suzuki Groups

We compute the Möbius function for the subgroup lattice of the simple Suzuki group $Sz(q)$; this is used to enumerate normal subgroups of certain Hecke groups with quotients isomorphic to $Sz(q)$.

亏群为Suzuki 2-群的块

Blocks whose defect groups are Suzuki $2$-groups

本研究完成了亏群为Suzuki 2-群的块的基本Morita等价分类，验证了Donovan猜想对这类群成立，并推导得到无非平凡奇阶正规子群的有限群的Suzuki Sylow 2-子群为平凡交的推论。

A Robinson characterization of finite $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $σ$-full if $G$ has a Hall $σ_{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ provided $G$ is $σ$-full and $A$ permutes with all Hall $σ_{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $σ$-permutability is a transitive relation in $G$, that is, if $K$ is a $σ$-permutable subgroup of $H$ and $H$ is a $σ$-permutable subgroup of $G$, then $K$ is a $σ$-permutabl…

On the sharp Baer--Suzuki theorem for $π$-radicals: sporadic groups

Let $π$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in the $π$-radical $\mathrm{O}_π(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to sporadic or alternating groups.

强4-拟正规性为传递关系的有限群

Finite groups in which strong 4-quasinormality is transitive relation

本文定义并研究强4-拟正规性具有传递性的有限Sq4T-群，证明这类群必可解，并补充4-可置换性相关结论与示例。

On the sharp Baer--Suzuki theorem for the $π$-radical

Let $π$ be a set of primes such that $|π|\geqslant 2$ and $π$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Oπ(G)$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.

具有单个对合共轭类的有限群的西洛2-子群

Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions

本文解决Brauer问题33，确定具有单个对合共轭类的有限群的西洛2-子群，除表1中非阿贝尔单群对应的子群外，还包括同循环群等，用到有限单群分类。

Ree群和Suzuki群的非离散欧氏建筑

Non-discrete Euclidean Buildings for the Ree and Suzuki groups

我们称一个非离散的欧几里得建筑物为Bruhat-Tits空间，如果它的自同构群包含一个子群，该子群诱导由该建筑物在无穷远处的根数据的所有根群生成的子群。这是一类非离散欧几里德建筑介绍和研究布鲁哈特和山雀。本文给出了Bruhat-Tits空间的完全分类，其无穷远处的建筑物是与具有通常根数据的Ree或Suzuki群相关联的B_2、F_4或G_2型周围建筑物的极性的不动点集。(In在B_2和G_2情形下，这个不动点集是秩为1的建筑物，在F_4情形下，它是一个广义八边形，其外尔群不是晶体学的。我们还表明，这些Bruhat-Tits空间中的每一个都有一个自然的嵌入在唯一的Bruhat-Tits空间，其建筑在无穷远是相应的环境建筑。

有限群Sylow子群的极大超群研究

On the maximal overgroups of Sylow subgroups of finite groups

该研究确定了Sylow r-子群含于唯一极大子群的有限群，通过约化到殆单群完成证明，推广了Aschbacher关于r=2的早期结果，并给出弱次正规子群新结果以研究Baer-Suzuki定理变体。

有限群可由一个π-子群和一个π'-子群生成

Finite groups can be generated by a pi-subgroup and a pi'-subgroup

该研究回答了Dan Haran的问题，推广了前人相关结果，证明任意有限群可由一个π-子群和一个π'-子群生成，并给出可数生成自由投射有限群的自由积描述。

Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $Γ$. If $G$ acts primitively on the points of $Γ$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group of Lie type. We show that, furthermore, the minimal normal subgroup $S$ of $G$ cannot be a Suzuki group or a Ree group of type $^2\text{G}_2$, and that if $S$ is a Ree group of type $^2\text{F}_4$, then $Γ$ is (up to point--line duality) the classical Ree--Tits generalised octagon.

The subgroup commutativity degree of finite P-groups

The subgroup commutativity degree of a group G has been defined in [6] as the probability that two subgroups of G commute, or equivalently that the product of two subgroups is again a subgroup. Problem 4.3 of [6] asks whether there exist families of groups other than dihedral, quasi-dihedral or generalized quaternion (all of 2-power cardinality), whose subgroup commutativity degree tends to 0 as the size of the group tends to infinity. An affirmative answer to this question has been provided by Aivazidis [1, 2] for the family of projective special linear groups over fields of even characterist…

关于有限群σ-性质理论的一个贡献

A contribution to the theory of $σ$-properties of a finite group

本文通过引入正规子群D避开主因子的条件，分别给出了有限群超可解性与有限可解群为PST-群的充要刻画，推进了有限群σ-性质理论。

无限群的反链条件

An antichain condition for infinite groups

本文研究了无限群的反链条件，证明了在特定子群性质下，广义radical群满足反链条件当且仅当满足RCC，并给出了相关群类的特征化。

Characterizations of some classes of finite $σ$-soluble $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $σ$-soluble if every chief factor $H/K$ of $G$ is a $σ_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i \in I$ such that $σ_{i}\cap π(G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ if $G$ has a complete Hall $σ$-…

特定 q 值下 PSL(2,q)、J1 和 Sz(q) 中正规原性子群的分类及格性质

Classification and lattice properties of pronormal subgroups in PSL(2,q), J1, and Sz(q) for the specified values of q

本文在特定 q 范围内完成了 PSL(2,q)、Sz(q) 和 J1 中正规原性子群的分类，并证明该子群族在并运算下封闭，在交运算下不封闭，但通过替换交运算可构成格。

关于abelized Johnson核的torsion子群非平凡性的研究

On the non-triviality of the torsion subgroup of the abelianized Johnson kernel

本文通过二维方法证明了abelized Johnson核的torsion子群非平凡，并给出了其基数的下界。

有限群的σ-性质的多项式时间判定

$σ$-properties of finite groups in polynomial time

本文证明了对置换群商群G/K的σ-幂零性、σ-可解性及子群商H/K的σ-次正规性、σ-p-可置换性均可在多项式时间内判定，并能求出相应的最小划分σ。

通过谱不变量研究因子分解数与子群交换度

Factorization number and subgroup commutativity degree via spectral invariants

本文利用子群不可置换图的邻接矩阵与拉普拉斯矩阵建立新的组合公式，从而用谱不变量表示有限群的因子分解数和子群交换度。

Bounding group orders by large character degrees: A question of Snyder

Let $G$ be a nonabelian finite group and let $d$ be an irreducible character degree of $G$. Then there is a positive integer $e$ so that $|G| = d(d+e)$. Snyder has shown that if $e > 1$, then $|G|$ is bounded by a function of $e$. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper, we will show for groups that have a nontrivial, abelian normal subgroup that $|G| \le e^4 - e^3$. We use this to prove that $|G| < e^4 + e^3$ for all groups. Given that there are a number of solvable groups that meet the first bound, it is best possible. Our work makes use of results rega…

交换子的阶与共轭类乘积在有限群中的研究

Orders of commutators and Products of conjugacy classes in finite groups

本文研究了有限群中交换子的阶与共轭类乘积的性质，证明了在特定条件下共轭类生成的子群是可解的。

Non-existence of integral Hopf orders for twists of several simple groups of Lie type

Let $p$ be a prime number and $q=p^m$, with $m \geq 1$ if $p \neq 2,3$ and $m>1$ otherwise. Let $Ω$ be any non-trivial twist for the complex group algebra of $\mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $\mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(\mathbb{C} \mathbf{PSL}_2(q))_Ω$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki groups, and for $\mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit …