A characterization of the groups $PSL_n(q)$ and $PSU_n(q)$ by their $2$-fusion systems, $q$ odd

Chiral polyhedra and finite simple groups

We prove that every finite non-abelian simple group acts as the automorphism group of a chiral polyhedron, apart from the groups $PSL_2(q)$, $PSL_3(q)$, $PSU_3(q)$ and $A_7$.

论李型例外群的中等秩李本原与极大子群

On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type

本文研究特征p李型群H嵌入例外代数群G的李本原子群，排除PSL_2型，证明除少数例外（如E_8中的若干小群）外不存在，并推广到自同构稳定性和极大子群刻画，结合表示论与计算方法。

Properties of element orders in covers for L(n,q) and U(n,q)

We show that if a finite simple group G isomorphic to PSL(n,q) or PSU(n,q), where either $n\ne 4$, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product contains an element whose order is distinct from every element order of G. As a consequence, we prove that the group PSL(n,q), where $n\ne 4$ or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka notebook.

Automorphisms of nonsplit extensions of 2-groups by $PSL_2(q)$

We complete the description of automorphism groups of all nonsplit extensions of elementary abelian $2$-groups by $PSL_2(q)$, with $q$ odd, for an irreducible induced action. An application of this result to the theory of $π$-submaximal subgroups is given.

Orbits of $\QQ^*(\sqrt{n})$ under the action of Modular Group $PSL(2,\mathbb {Z})$

Coset diagrams have been used to study quotients, orbits, subgroups and structure of the finitely generated groups. In this paper we use coset diagrams and modular arithmetic to determine the $G$-orbits of $\QQ^*(\sqrt{p^k})$, $\QQ^*(\sqrt{2p^k})$, $\QQ^*(\sqrt{2^2p^k})$, and in general $\QQ^*(\sqrt{2^lp^k})$, for each $l\geq3$ and $k=2h+1\geq3$, for each odd prime $p$.

$θ$-semisimple twisted conjugacy classes of type D in $\operatorname{PSL}_n(q)$

Let $p$ be an odd prime, $m\in {\mathbb N}$ and set $q=p^m$, $G=\operatorname{PSL}_n(q)$. Let $θ$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and $\operatorname{Fr}_q$ be the Frobenius endomorphism of $\operatorname{PSL}_n(\overline{{\mathbb F}_q})$. We show that every $(d\circ θ)$-conjugacy class of a $(d\circ θ,p)$-regular element in $G$ is represented in some $\operatorname{Fr}_q$-stable maximal torus and that most of them are of type D. We write out the possible exceptions and show that, in particular, if $n\geq5$ is either odd or a multiple of $4$ and $q>7$, th…

Groups of prime degree and the Bateman-Horn Conjecture

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We present heuristic arguments and computational evidence based on the Bateman-Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\rm PSL}_n(q)$, ${\rm PSU}_n(q)$ and ${\rm PSp}_{2n…

Cubic graphical regular representations of some classical simple groups

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}Ω_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}Ω_{n}^{-}(q)$ ($n=8, 10, 12$), where $q = 2^f$ with $f \ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q \rightarrow \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^{i}-1$ for each $1 \le i < ef$ t…

Quasisimple classical groups and their complex group algebras

Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and $S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that, excluding the cases when the simple group $S$ has a very exceptional Schur multiplier such as $\PSL_3(4)$ or $\PSU_4(3)$, $H$ is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.

Moufang sets of finite Morley rank of odd type

We show that for a wide class of groups of finite Morley rank the presence of a split $BN$-pair of Tits rank $1$ forces the group to be of the form $\operatorname{PSL}_2$ and the $BN$-pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian $2$-subgroups and show that all such Moufang sets are standard (and thus associated to $\operatorname{PSL}_2(F)$ for $F$ an algebraically closed field of chara…

On the diameters of McKay graphs for finite simple groups

Let $G$ be a finite group, and $α$ a nontrivial character of $G$. The McKay graph ${\mathcal M}(G,α)$ has the irreducible characters of $G$ as vertices, with an edge from $χ_1$ to $χ_2$ if $χ_2$ is a constituent of $αχ_1$. We study the diameters of McKay graphs for simple groups $G$. For $G$ a group of Lie type, we show that for any $α$, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for $G={\rm PSL}_n(q)$ or ${\rm PSU}_n(q)$. We also bound the diameter for symmetric and alternating groups.

Adjoint representations of black box groups ${\rm PSL}_2(\mathbb{F}_q)$

Given a black box group $\mathsf{Y}$ encrypting $\rm{PSL}_2(\mathbb{F})$ over an unknown field $\mathbb{F}$ of unknown odd characteristic $p$ and a global exponent $E$ for $\mathsf{Y}$ (that is, an integer $E$ such that $\mathsf{y}^E=1$ for all $\mathsf{y} \in \mathsf{Y}$), we present a Las Vegas algorithm which constructs a unipotent element in $\mathsf{Y}$. The running time of our algorithm is polynomial in $\log E$. This answers the question posed by Babai and Beals in 1999. We also find the characteristic of the underlying field in time polynomial in $\log E$ and linear in $p$. Furthermore…

有限群中可解化子的数量

The Number of Solvabilizers in Finite Groups

本文研究了有限群中可解化子数量的计算方法，针对不同类型的非阿贝尔单群给出了具体结果，并提出了一种适用于更广泛群类的算法。

通过元素阶乘积的素因子分解中的指数集合进行识别

Recognition by the set of exponents in the prime factorization of the product of element orders

本文利用元素阶乘积的素因子分解指数集合，证明了若干射影特殊线性群及循环群可由该指数集合（或结合群阶）唯一确定。

On the Surjectivity of Engel Words on PSL(2,q)

We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evid…

Natural representations of black box groups encrypting $SL_2(\mathbb{F}_q)$

Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\rm SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in $\log E$, the isomorphisms \[ \mathsf{Y} \longleftrightarrow {\rm SL}_2(\mathsf{K}), \] where $\mathsf{K}$ is a black box field encrypting $\mathbb{F}$. Our algorithm makes no reference to any additional oracles. We also give similar algorithms for black box groups encrypting ${\rm PGL}_2(\mathbb{F})$, ${\rm PSL}_2(\mathbb{F})$.

On CT and CSA Groups and Related Ideas

A group is $G$ commutative transitive or CT if commuting is transitive on nontrivial elements. A group $G$ is CSA or conjugately separated abelian if maximal abelian subgroups are malnormal. These concepts have played a prominent role in the studies of fully residually free groups, limit groups and dicriminating groups. They were especially important in the solution to the Tarski problems. CSA always implies CT however the class of CSA groups is a proper subclass of the class of CT groups. For limit groups and finitely generated elementary free groups they are equivalent. In this paper we exam…

On some rational extension properties for $GL_n(q)$ and even-degree characters fixed by order-2 Galois automorphisms

In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.

Non-solvable groups whose character degree graph has a cut-vertex. II

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $Δ(G)$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper continues the work, started in [7], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected compone…

具有负素数幂欧拉示性数的不可定向正则地图

Non-orientable regular maps with negative prime-power Euler characteristic

本文对欧拉示性数为负奇素数幂的曲面上的正则地图进行分类，通过分析自同构群商群结构，得到18个非空正则地图族。

扭本原群结合方案

Twisted primitive group association schemes

本研究针对本原群结合方案的相交数是否决定其组合同构类的问题，构造了多类与群共轭类分划代数同构但组合不同构的Schur分划，证明相关本原群结合方案的相交数无法确定其组合同构类。

关于有限群的幂零图

On the Nilpotent Graph of a finite Group

本文研究有限非幂零群的幂零图，给出团数与连通分支数的下界，并确定对称群、二面体群及 PSL(2,q) 的连通分支数及相关图论性质。

关于图、群与几何

On Graphs, Groups and Geometry

该研究基于Leemann与de la Salle的Cayley图刚性理论，定义自然群并刻画了连通李群、有限生成群等各类群的自然性判定条件，给出自然与非自然群的实例。