An Atlas of subgroup lattices of finite almost simple groups

特征标与几乎单群的Sylow 3-子群的生成

Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups

本研究针对几乎单群，证明其特征标表可算法判定Sylow 3-子群是否2生成，且该性质等价于主块特征标的Galois作用条件，是Alperin-McKay-Navarro猜想的推论。

Subgroups of almost finitely presented groups

We show that every countable group embeds in a group of type $FP_2$.

The geometry of groups containing almost normal subgroups

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=\infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a fini…

关于几乎单群和本原完全群的最大子群

On the maximal subgroups of almost simple and primitive perfect groups

本文证明有限几乎单群的最大子群的导出列第10项是完全群，并推广到完全群的无核最大子群，且常数10是最优的。

Generating maximal subgroups of finite almost simple groups

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

几乎单经典群的大极大子群

Large maximal subgroups of almost simple classical groups

本文刻画以有限单经典群为基座的几乎单群的大极大子群，确定了相关几乎单群集合非空的充要条件。

Almost Abelian Lie groups, subgroups and quotients

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified a…

Maximal subgroups of small index of finite almost simple groups

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.

关于 $E_7(q)$ 的极大子群及相关几乎单群

On the Maximal Subgroups of $E_7(q)$ and Related Almost Simple Groups

本文几乎完全分类了 $E_7(q)$ 的极大子群，仅缺失四个特殊情形，并完整确定了 $q$ 为 $3$ 的幂及 $q=4$ 时的极大子群。

Abelian群中极大几乎周期子群的极大几乎周期子群

Maximally almost periodic subgroups of elementary Abelian topological groups

本文研究了无限Abelian拓扑群的极大几乎周期子群，证明了任何无限Abelian拓扑群的素数指数都具有无限的极大几乎周期子群。

Groups where free subgroups are abundant

Given an infinite topological group G and a cardinal k>0, we say that G is almost k-free if the set of k-tuples in G^k which freely generate free subgroups of G is dense in G^k. In this note we examine groups having this property and construct examples. For instance, we show that if G is a non-discrete Hausdorff topological group that contains a dense free subgroup of rank k>0, then G is almost k-free. A consequence of this is that for any infinite set X, the group of all permutations of X is almost 2^|X|-free. We also show that an infinite topological group is almost aleph_0-free if and only …

极大子群的指数几乎为奇数的有限群

Finite groups whose maximal subgroups have almost odd index

本文研究了由群不变量的算术性质决定有限群结构的问题，确定了极大子群的指数不被4整除的有限群的结构，并由此得到了一些新的2-幂零性准则。

例外群 $F_4(q)$、$E_6(q)$ 和 ${}^2E_6(q)$ 及相关几乎单群的极大子群

The maximal subgroups of the exceptional groups $F_4(q)$, $E_6(q)$ and ${}^2E_6(q)$ and related almost simple groups

本文综合代数群、表示论、计算代数与三线性型方法，完整确定所有有限域上 $F_4(q)$、$E_6(q)$、${}^2E_6(q)$ 及相关几乎单群的极大子群，并更正 ${}^2F_4(8)$ 的列表。

Second maximal subgroups of the finite alternating and symmetric groups

A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the maximal subgroups that contain a given second maximal subgroup in some finite group G? Various reduction arguments are available except when G is almost simple. Following the classification of the finite simple groups, finite almost simple groups fall into three categories: alternating and symmetric groups, almost simple groups of Lie type, sporadic groups an…

确实，Monster群没有基柱为$\text{PSL}_2(16)$的几乎单极大子群

Indeed, the Monster has no almost simple maximal subgroup with socle $\text{PSL}_2(16)$

该研究证明了Monster群不存在基柱为$\text{PSL}_2(16)$的几乎单极大子群，验证了Holmes与威尔逊未发表结果中的对应情形，证明基于mmgroup包的可复现计算，剩余两种情形将在扩展版中处理。

Groups which are almost groups of Lie type in characteristic p

For a prime $p$, a $p$-subgroup of a finite group $G$ is said to be large if and only if $Q= F^*(N_G(Q))$ and, for all $1 \neq U \le Z(Q)$, $N_G(U) \le N_G(Q)$. In this article we determine those groups $G$ which have a large subgroup and which in addition have a proper subgroup $H$ containing a Sylow $p$-subgroup of $G$ with $F^*(H)$ a group of Lie type in characteristic $p$ and rank at least 2 (excluding $\PSL_3(p^a)$) and $C_H(z)$ soluble for some $z \in Z(S)$. This work is part of a project to determine the groups $G$ which contain a large $p$-subgroup.

On solvable factors of almost simple groups

Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Li and Xia describe the factorizations of almost simple groups with a solvable factor $H$. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of $G_0$ containing $H \cap G_0$ is iden…

Unimodularity of Invariant Random Subgroups

An invariant random subgroup $H \leq G$ is a random closed subgroup whose law is invariant to conjugation by all elements of $G$. When $G$ is locally compact and second countable, we show that for every invariant random subgroup $H \leq G$ there almost surely exists an invariant measure on $G/H$. Equivalently, the modular function of $H$ is almost surely equal to the modular function of $G$, restricted to $H$. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact…

Hyperbolic groups with almost finitely presented subgroups

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.

具有交错、古典或零散基座的几乎单群的第二极大子群的结构与生成

The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle

针对基座为交错、古典或零散群的几乎单群，修正了其非抛物极大子群生成元数量的分类，将生成元界优化至最优值，确定了相关主因子并改进了已有结论。

Carter subgroups of finite groups

In the paper it is proven that Carter subgroups of a finite group are conjugate. A complete classification of Carter subgroups in finite almost simple groups is also obtained.

Weakly commensurable S-arithmetic subgroups in almost simple algebraic groups of types B and C

Let G and G' be absolutely almost simple algebraic groups of types B and C respectively, of rank at least 3, and defined over a number field K. We determine when G and G' have the same isomorphism or isogeny classes of maximal K-tori. This leads to the necessary and sufficient conditions for two Zariski-dense S-arithmetic subgroups of G and G' to be weakly commensurable.

On Non-Generic Finite Subgroups of Exceptional Algebraic Groups

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric…

Alternating Subgroups of Exceptional Groups of Lie Type

In this paper we examine embeddings of alternating groups and symmetric groups into almost simple groups of exceptional type. In particular, we prove that unless the alternating or symmetric group has degree 6 or 7, there is no maximal subgroup of any almost simple group with socle an exceptional group of Lie type that is an alternating or symmetric group. Furthermore, in the remaining open cases we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aimi…

Maximal $\mathrm{PSL_2}$ subgroups of exceptional groups of Lie type

We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group. Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves t…

On the intersection of solvable Hall subgroups in finite simple exceptional groups of Lie type

Assume that a finite almost simple group with simple socle isomorphic to an exceptional group of Lie type possesses a solvable Hall subgroup. Then there exist four conjugates of the subgroup such that their intersection is trivial.

全形包含可解正则子群的有限几乎单群

Finite almost simple groups whose holomorph contains a solvable regular subgroup

本文分类了全形包含可解正则子群的有限几乎单群，推广了先前对非交换单群的结果，并刻画了可解扩张上的Hopf-Galois结构类型。

Irreducible almost simple subgroups of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V) of this form, where $V \ne W, W^{*}$ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible …

Finite quasiprimitive permutation groups with a metacyclic transitive subgroup

In this paper, we classify finite quasiprimitive permutation groups with a metacyclic transitive subgroup, solving a problem initiated by Wielandt in 1949. It also involves the classification of factorizations of almost simple groups with a metacyclic factor.

A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups

The aim of this brief note is to offer another proof of a theorem of Hrushovski that approximate subgroups of GL_n(C) are almost nilpotent. This approach generalizes to uniformly non amenable groups.

Intersection of conjugate solvable subgroups in finite classical groups

We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five conjugates of $H$ whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a finite group. In particular, a positive answer to Vdovin's problem yields that if $G$ has a solvable subgroup of index $n$, then it has a solvable normal subgroup of …

The isomorphism problem for almost split Kac-Moody groups

We consider the isomorphism problem for almost split Kac--Moody groups, which have been constructed by Rémy via Galois descent from split Kac-Moody groups as defined by Tits. We show that under certain technical assumptions, any isomorphism between two such groups must preserve the canonical subgroup structure, i.e. the twin root datum associated to these groups, which generalizes results of Caprace in the split case. An important technical tool we use is the existence of maximal split subgroups inside almost split Kac-Moody groups, which generalizes the corresponding result of Borel-Tits for …

Centralizers of Finite Subgroups of the Mapping Class Group

In this paper, we study the action of finite subgroups of the mapping class group of a surface on the curve complex. We prove that if the diameter of the almost fixed point set of a finite subgroup H is big enough, then the centralizer of H is infinite.

两个子群何时是独立的？

When are Two Subgroups Independent?

本文提出子群独立的新定义，指出传统几乎不相交条件不足，给出了必要和充分条件，并提出算法用于判断子群独立性。

Intersection of conjugate solvable subgroups in symmetric groups

It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$

On the number of classes of conjugate Hall subgroups in finite simple groups

In this paper we find the number of conjugate $π$-Hall subgroups in all finite almost simple groups. We also complete the classification of $π$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.

魔群（Monster）的极大子群

The maximal subgroups of the Monster

本文利用Python包mmgroup的可复现计算，解决了魔群极大子群分类的遗留问题，发现两个新极大子群，纠正了此前的错误论断，完成了全部散在单群的极大子群分类。

Mixing actions of countable groups are almost free

A measure preserving action of a countably infinite group Γis called totally ergodic if every infinite subgroup of Γacts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γis totally ergodic then there exists a finite normal subgroup N of Γsuch that the stabilizer of almost every point is equal to N. Surprisingly the proof relies on the group theoretic fact (proved by Hall and Kulatilaka as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely o…

On parabolic subgroups of Artin-Tits groups

Abstract. We address the conjecture which states that an intersection of parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove that the conjecture is equivalent to a, a priori, weaker conjecture. We also prove the conjecture in a specific case. Along the way, we provide short and almost self-contain algebraical proofs of several classical results on Artin-Tits groups, such as those of Van der Lek on intersection of standard parabolic subgroups.

关于几乎单群中西罗子群的交集

On the intersections of Sylow subgroups in almost simple groups

针对有限几乎单群中西罗子群交集的性质，本文采用基于不动点比估计的概率方法，完成了Zenkov关于西罗子群交集为平凡子群的分类工作。

Groups of cohomological codimension one

We show that if $H$ is an almost normal subgroup of $G$ such that both $H$ and $G$ are of type $VFP$ and vcd($G$) = vcd($H$) + 1, then $G$ is the fundamental group of a graph of groups in which all vertex and edge groups are commensurable to $H$. We also investigate almost normal subgroups of one-relator groups and duality groups.

Raikov完备的几乎可度量化正规子群所得商群的Raikov余项

Raikov Remainders of Quotients by Raikov-Complete Almost Metrizable Normal Subgroups

本文研究拓扑群商群的Raikov余项性质，刻画了商同态延拓下余项等式成立的充要条件，证明Raikov完备几乎可度量化闭正规子群满足该条件，得出伪紧性可传递的结论。

Group laws and free subgroups in topological groups

We prove that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property we show that almost all finite subsets of the group generate free subgroups. We derive consequences of these theorems on Thompson's group $F$, weakly branch groups, automorphism groups of regular trees and profinite groups with alternating composition factors of unbounded degree.

Maximal subgroups of sporadic groups

A systematic study of maximal subgroups of the sporadic simple groups began in the 1960s. The work is now almost complete, only a few cases in the Monster remaining outstanding. We give a survey of results obtained, and methods used, over the past 50 years, for the classification of maximal subgroups of sporadic simple groups, and their automorphism groups.

强子群递归与Nevo-Stuck-Zimmer定理

Strong subgroup recurrence and the Nevo-Stuck-Zimmer theorem

本文提出了可数群Chabauty空间中的回旋镖子群概念，证明了高秩绝对几乎单群的S-算术群只有极少的此类子群，从而恢复了Margulis正规子群定理与Nevo-Stuck-Zimmer定理。

Almost group theory

The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\widetilde{\mathfrak M}\_c$-groups.

Invariant random subgroups of linear groups

Let $Γ< \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(Γ)$ the space of all subgroups of $Γ$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $Γ$ is a conjugation invariant Borel probability measure on $\mathrm{Sub}(Γ)$. An $\mathrm{IRS}$ is called nontrivial if it does not have an atom in the trivial group, i.e. if it is nontrivial almost surely. We denote by $\mathrm{IRS}^{0}(Γ)$ the collection of all nontrivial $\mathrm{IRS}$ on $Γ$. We show that there exits a free subgroup $F < Γ$ and a…

Automorphic orbits in free groups: words versus subgroups

We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F_2? Moreover, decidability subsists if we allow H to be a rational subset of F_2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, we show the decidability of the following weaker problem: given a finitely generated subgroup H, a word u and an integer k, d…

Thompson's group $F$ is almost $\frac{3}{2}$-generated

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element of $G$ belongs to a generating pair of $G$. Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. Recently, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. In this paper, we prove that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part…

The fundamental group of the $p$-subgroup complex

We study the fundamental group of the $p$-subgroup complex of a finite group $G$. We show first that $π_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ is the alternating group on $10$ letters). This is the first concrete example in the literature of a $p$-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of M. Aschbacher, $π_1(A_p(G)) = π_1(A_p(S_G)) * F$, where $F$ is a free group and $π_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G = Ω_1(G)/O_{p'}(Ω_1(G))$. This result essentially reduces the study of the fundamental group of …

Silhouettes and generic properties of subgroups of the modular group

We show how to count and randomly generate finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$ of a given isomorphism type. We also prove that almost malnormality and non-parabolicity are negligible properties for these subgroups. The combinatorial methods developed to achieve these results bring to light a natural map, which associates with any finitely generated subgroup of $\textsf{PSL}(2,\mathbb{Z})$ a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of $\textsf{PSL}(2,\mathbb{Z})$.

连通李群自同构群的几乎代数性

On the almost algebraicity of groups of automorphisms of connected Lie groups

本文研究连通李群自同构群的几乎代数性，在$C$可能非平凡时给出充要条件，对特定李群类得到更具体判据，还分析了相关子群的几乎代数性。

Bernoulli disjointness and maximally almost periodic groups

We show that any discrete group containing an infinite, normal, maximally almost periodic subgroup has the Bernoulli Disjointness Property, or BDJ. Also, any group containing enough infinite normal subgroups to separate points has the BDJ. Lastly, any group admitting a minimal free proximal flow has the BDJ.

Number of Sylow subgroups in finite groups

Denote by $ν_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ν_p(H)\leqν_p(G)$ for $H\leq G$, however $ν_p(H)$ does not divide $ν_p(G)$ in general. In this paper we reduce the question whether $ν_p(H)$ divides $ν_p(G)$ for every $H\leq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.