Michael Giudici
We determine all maximal subgroups of the almost simple groups with socle $T=\PSL(2,q)$, that is, of all groups $G$ such that $\PSL(2,q)\leqslant G\leqslant\PGammaL(2,q)$, with $q\geq 4$.
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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。
2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q)
本文研究了允许特定自同构群的2-(v,k,3)设计的分类问题，重点在于其socle结构为PSL(2,q)的情况。
Block-transitive $t$-($k^2,k,λ$) designs with $PSL(n,q)$ as socle
针对以PSL(n,q)（n≥3）为基座的非平凡区组传递t-(k²,k,λ)设计开展研究，证明t=2并确定其全部可能参数，同时给出λ整除k时的λ取值范围。
Let $\mathcal D$ be a nontrivial symmetric $(v,k,λ)$ design, and $G$ be a subgroup of the full automorphism group of $\mathcal D$. In this paper we prove that if $G$ acts flag-transitively, point-primitively on $\mathcal D$ and $Soc(G)= PSL(2,q)$, then D has parameters $(7, 3, 1)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 3)$ or $(15, 8, 4)$.
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…
The socle of a group $G$ is the subgroup generated by all minimal normal subgroups of $G$. In this short note, we determine the socle of a Hamiltonian group explicitly.
$2$-designs admitting a flag-transitive automorphism group with socle $PSL(2,q)$
研究了具有基座为\(PSL(2,q)\)（\(q = p^{f}\geq 4\)）的旗传递自同构群的\(2 -\)设计，分点本原和点非本原情况，后者获完全分类，前者除特定线性空间外还分类出\(48\)个零星例子。
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。
The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$
本文证明了对于socle为Ree(q)或Sz(q)的原始群，其Saxl图中任意两个顶点共享共同邻居的猜想。
We prove that the Monster does not contain any subgroup isomorphic to PSL_2(27).
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。
In this paper we characterize the elements of PSL(2,Z), as a subgroup of Thompson group T, in the language of reduced tree pair diagrams and in terms of piecewise linear maps as well. Actually, we construct the reduced tree pair diagram for every element of PSL(2,Z) in normal form. This allows us to estimate the length of the elements of PSL(2,Z) through the number of carets of their reduced tree pair diagrams and, as a consequence, to prove that PSL(2,Z) is a non distorted subgroup of T. In particular, we find non-distorted free non abelian subgroups of T.
The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q)
本文研究了原始群中socle为PSU(3,q)的情况，验证了Burness-Giudici猜想的大部分情况，排除了包含PSO(3,q)的特殊情况。
The Burness-Giudici Conjecture on Primitive Groups with Socle PSU(3,q)
本文证明了Socle为PSU(3,q)的原始群满足Saxl图中任意两个顶点共享公共邻居的猜想。
Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{G}^σ$ its fixed-point subgroup under the action of a Steinberg endomorphism $σ$, and $\overline{G}$ an almost simple group with socle $G$. A maximal subgroup $M<\overline{G}$ is called non-generic if it is almost simple and its socle is not isomorphic to a group of Lie type in characteristic $p$. A finite subgroup $H$ of $\boldsymbol{G}$ is Lie primitive if it does not lie in any proper closed positive-dimensional subgroup of $\boldsymbol{G}$; it is Lie imprimitive if it lies in a positive-…
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。
Transitive sets of derangements in primitive actions of PSL_2(q)
本文研究了PSL_2(q)在原始作用中的传递错位集，证明此类群不构成负例。
On subgroups of the Monster isomorphic to $PSL_2(8)$
本文通过计算机计算分类 Monster 中包含 $7B$-元素的 $PSL_2(8)$ 子群，证明其不存在，并验证了早期及独立计算的可复现性。
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。
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 $\textsf{PSL}(2,\mathbb{Z})$.
While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider subgroups $Γ$ of arithmetic groups in $PSL(2,C)^q \times PSL(2,R)^r$ with $q+r>1$ and their limit set. We prove that the projective limit set of a nonelementary finitely generated $Γ$ consists of exactly one point if and only if one and hence all projections of $Γ$ to the simple factors of $PSL(2,C)^q \times PSL(2,R)^r$ are subgroups of arithmetic Fuchsian or Kleinia…
Let $G$ be a finite group and $cd(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is Mathieu group such that $cd(G) =cd(H)$, then there exists an Abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. This study is heading towards the study of an extension of Huppert's conjecture (2000) for almost simple groups.
We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with a "big" limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups admitting modular embeddings.
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.
In this paper we give a description of the possible limit sets of finitely generated subgroups of irreducible lattices in $PSL(2,R)^r$.
We show that the group $PSL(2,R)$ is a spherical subgroup in the group of $C^3$-diffeomorphisms of the circle. Also, the group of automorphisms of a Bruhat--Tits tree is a spherical subgroup in the group of hierarchomorphisms of the tree.
Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup H<G is closed in the pro-normal topology. As a corollary we deduce that if M is a maximal subgroup of a lattice in PSL(2,C) then either M is finite index or M is not finitely generated.
In this short note we prove that the finite non-abelian simple groups PSL(2,q), where q = 5,7, are determined by their posets of classes of isomorphic subgroups. In particular, this disproves the conjecture in the end of [5].
We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to $\mathrm{PSL}_2(q)$, ${^2\mathrm{B}_2}(q)$, ${^2\mathrm{G}_2}(q)$ or $\mathrm{PSU}_3(q)$. Our method uses the notion of a "strongly non-binary action".
This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n \geq 3. Alongside this analysis, we give a construction for a flag-transitive 2-(v,k-1,k-2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line.…
Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is a sporadic simple group H0 such that cd(G) = cd(H), then G' = H0 and there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert's conjecture (2000), we also provide some examples to show that G is not necessarily a direct product of A and H, and hence we cannot extend this conjecture to almost simple groups.
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.
We analyze the group structures of two groups of order 1344 which are respectively non-split and split extensions of the elementary Abelian group of order 8 by its automorphism group PSL_2(7).They share the same character table. The group 2^3.PSL_2(7) is a finite subgroup of the Lie Group G_2 preserving the set of octonions \pm e_i , (i=1,2,...,7) representing a 7-dimensional octahedron.Its three maximal subgroups 2^3:7:3, 2^3.S_4 and 4.S_4:2 correspond to the finite subgroups of the Lie groups G_2, SO(4) and SU(3) respectively. The group 2^3:PSL_2(7) representing the split extension possesses…
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 $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$
We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an almost 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first posit…
We show that for every prime $r$ all $r$-subgroups in the normalized units of the integral group ring of $\operatorname{PSL}(2,p^3)$ are isomorphic to subgroups of $\operatorname{PSL}(2,p^3)$. This answers a question of M. Hertweck, C.R. Höfert and W. Kimmerle for this series of groups.
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。
On computing finite index subgroups of PSL(2,Z)
本文基于 Kulkarni 思想提出递归计算二价树及自同构群的方法，用于枚举 $PSL(2,\mathbb{Z})$ 中指数不超过 20 的全部子群，并整理其算术性质。
Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $Φ(H)\neq H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $Φ(H)$ in $\mathrm{PSL}_2(\mathbb{R})$ is discrete. We thus obtain a natural family of thin subgroups of $\mathrm{PSL}_2(\mathbb{R})$ whose commensurators are discrete, establishing some cases of a conjecture of Shalom.
Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\mathbb{Z}G)$ is isomorphic to a subgroup of $G$.
Let $G$ be the simple group ${\rm PSL}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the Möbius function of $H$ in the subgroup lattice of $G$. To this aim, we describe the intersections of maximal subgroups of $G$. We point out some connections of the Möbius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of $r$-subgroups of ${\rm PGL}(3,q)$, for any prime $r$ and any prime power $q$.
Lifting subgroups of $\mathrm{PSL}_2$ to $\mathrm{SL}_2$ over local fields
本文研究非阿基米德局部域上从 PSL₂(K) 到 SL₂(K) 的子群提升问题，证明无 2-挠的离散子群总能提升，并举例说明两个条件缺一不可，同时简要讨论表示提升。
It is shown that the number $f_λ$ of free subgroups of index $6λ$ in the modular group $\PSL_2(\Z)$, when considered modulo a prime power $p^\al$ with $p\ge5$, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing $\PSL_2(\Z)$ as a special case), and for a one-parameter family of lifts of the Hecke group $\mathfrak{H}(4)=C_2*C_4$. All this is achieved by explicitly determining Padé approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complemen…
Conjugacy class fusion from four maximal subgroups of the Monster
本文确定了 Monster 群的四个极大子群到 Monster 群的共轭类融合，以证明将这些数据添加到计算代数系统 GAP 的 Character Table Library 中是合理的，并通过 Python 包 mmgroup 的可重现计算支持了相关证明。
Elementary groups in $\PSL(3,\C)$
本文分类了 $\PSL(3,\mathbb{C})$ 中 Kulkarni 极限集由有限多条一般位置直线构成的初等子群，揭示了这类群的结构特征。
The problem of determining whether or not a non-elementary subgroup of $PSL(2,\CC)$ is discrete is a long standing one. The importance of two generator subgroups comes from Jørgensen's inequality which has as a corollary the fact that a non-elementary subgroup of $PSL(2,\CC)$ is discrete if and only if every non-elementary two generator subgroup is. A solution even in the two-generator $PSL(2,\RR)$ case appears to require an algorithm that relies on a the concept of {\sl trace minimizing} that was initiated by Rosenberger and Purzitsky in the 1970's Their work has lead to many discreteness res…
Representations of Solvable Subgroups of PSL(3,C)
本文完整描述了PSL(3,C)中上三角复克莱因群的表示，确定了含双曲元素的16种类型，推进了初等离散子群的研究。
We describe the centralizer of irreducible representations from a finitely generated group $Γ$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $χ^i(Γ,PSL(p,\mathbb{C}))$. When $Γ$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of…
p-Nilpotent maximal subgroups in finite groups
该研究针对每个极大子群要么p-幂零要么有素指数的有限群，证明其仅可涉入PSL_n(q)族中一种阶被p整除的非交换单群，且p=2时该单群必为特定参数的PSL_2(r^a)。
We determine the semi-regular subgroups of the 2-transitive permutation groups PGL(2,n), PSL(2,n), PGU(3,n), PSU(3,n), Sz(n) and Ree(n) with n a suitable power of a prime number p.
We count the finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $Γ(H)$, we consider the number of vertices of $Γ(H)$ to be the size of $H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $\textsf{PSL}(2,\mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $\textsf{PSL}(2,\mathbb{Z})$, as well as of the number of finite …
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 $p$.
Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$
该研究推导了${\ m PSL}_2(p)$（p为素数）非单位真子群的同构类、共轭类及两类自正规相关共轭类的计数公式，证明了p>37时的四类下界，结合数值搜索与数论结果给出了下界可达性与上界结论。
The first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian transitive characteristically simple subgroups $H$ of finite quasiprimitive permutation groups $G$. If the socle of $G$, denoted by $\mbox{soc}(G)$, is nonabelian, then $H$ lies in $\mbox{soc}(G)$. An explicit description is given for the possibilities of $H$ under the condition that $H$ does not contain a nontrivial normal subgroup of $\mbox{soc}(G)$.
On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups
本文研究有限群谱识别问题，证明 Janko 群 J1 和小 Ree 群 2G2(q) 的直积平方可由谱唯一确定，而 PSL2(q) 的直积平方则存在无穷多个同谱群。
Discrete two-generator subgroups of ${\rm PSL_2}$ over non-archimedean local fields
本文研究非阿基米德局部域上 ${\rm PSL_2}$ 二元生成子群的离散性，给出充要条件与判定算法，并建立相关 ${\rm SL_2}$ 子群稠密性算法。
We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n,Z) which is not 2-transitive.
The length of mixed identities for finite groups
本文证明了存在常数 $c>0$，使得没有长度 $\leq c$ 的非平凡混合等式的有限群必为以 Lie 型单群为 socle 的几乎单群，并给出了以 ${\rm PSL}_n(q)$ 等为 socle 的群的最短非平凡混合等式长度仅依赖域大小 $q$ 的与秩无关的界。
For G = PSL(2,p^f) denote by ZG the integral group ring, by V(ZG) the group of normalized units of ZG and let r be a prime different from p. Using the so called HeLP-method we prove, that units of r-power order in V(ZG) are rationally conjugate to elements of G. As a consequence we prove, that subgroups of prime power order in V(ZG) are rationally conjugate to subgroups of G, provided p = 2 or f =1.
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…
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