$PSL(3,q)$ and line-transitive linear spaces

PSL_2(q)在原始作用中的传递错位集

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。

Classification of continuously transitive circle groups

Let G be a closed transitive subgroup of Homeo(S^1) which contains a non-constant continuous path f: [0,1] --> G. We show that up to conjugation G is one of the following groups: SO(2,R), PSL(2,R), PSL_k(2,R), Homeo_k(S^1), Homeo(S^1). This verifies the classification suggested by Ghys [Enseign. Math. 47 (2001) 329-407]. As a corollary we show that the group PSL(2,R) is a maximal closed subgroup of Homeo(S^1) (we understand this is a conjecture of de la Harpe). We also show that if such a group G < Homeo(S^1) acts continuously transitively on k-tuples of points, k>3, then the closure of G is H…

The generating pairs of the 2-transitive groups

Given a finite group $G$. The generating pair $(H,a)$ of $G$, that is, $H<G$ and $a\in G$ such that $\langle a,H\rangle=G$. In this paper, we introduce the definition of FF-subgroup to characterize the generating pairs of the symmetric groups, alternating groups and projective groups $PSL(2,q)$. This gives a partial answer to an open problem of J. André and J. Araújo and P. J. Cameron.

Finding conjugate stabilizer subgroups in PSL(2; q) and related groups

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…

Groupes d'isométries permutant doublement transitivement un ensemble de droites vectorielles

Let X be a non-empty finite set, E be a finite dimensional euclidean vector space and G a finite subgroup of O(E), the orthognal group of E. Suppose GG={U_i | i in X} is a finite set of linear lines in E and an orbit of G on which its operation is twice transitive. Then GG is an equiangular set of lines, which means that we can find a real number "c", and generators u_i of the lines U_i (i in X) such that forall i,j in X, ||u_i||=1, and if i is different from j then (u_i|u_j)=\gve_{i,j}.c, with \gve_{i,j} in {-1,+1\} Let Gamma be the simple graph whose set of vertices is X, two of them, say i …

与二维射影特殊线性群相关的区传递 t-(k^2,k,λ) 设计

Block-transitive t-(k^2,k,λ) designs associated to two dimensional projective special linear groups

本文研究区传递自同构群作用下的 t-(k^2,k,λ) 设计，证明当正规子群为 PSL(2,q)（q≥4）时，仅当 q=8 时存在非平凡设计，且必为参数 λ∈{2,6,9,12,18,36} 的 2-(36,6,λ) 设计。

推升型局部s-弧传递图的顶点稳定子

Vertex stabilizers of locally $s$-arc transitive graphs of pushing up type

本文研究推升型局部s-弧传递图（s≥4）的顶点稳定子结构，证明了O_p(G_x^{[1]})为初等阿贝尔群，且商群G_x/G_x^{[1]}同构于介于PSL_2(p^a)与PΓL_2(p^a)之间的群X。

Linear fractional group as Galois group

We compute all signatures of $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ which classify all orientation preserving actions of the groups $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ on compact, connected, orientable surfaces with orbifold genus $\geq 0$. This classification is well-grounded in the other branches of Mathematics like topology, smooth, and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.

v<100的旗传递且点非本原对称$(v,k,λ)$设计

The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,λ)$ Designs with $v<100$

本文完成v<100的旗传递点非本原对称2-(v,k,λ)设计的分类，新发现两个不同构的2-(64,28,12)设计，通过PSL₂(7)的绝对不可约8维𝔽₂-表示理论构造并证明其自同构群性质，结合已有结果实现v<100旗传递2-设计的完全分类。

Line-transitive point-imprimitive linear spaces with Fang-Li parameter gcd(k,r) at most 10

This paper is a further contribution to the classification of line-transitive finite linear spaces. We prove that if S is a non-trivial finite linear space with the Fang-Li parameter gcd(k,r) is 9 or 10, the automorphism group G of S is line-transitive and point-imprimitive, then S is the Desarguesian projective plane PG(2,9).

一个新的旗传递线性空间

A new flag-transitive linear space

该研究构造了具PΓL₂(32)自同构群的新旗传递2-(496,4,1)设计，修正了有限旗传递线性空间分类的遗漏。

Projective linear groups as automorphism groups of chiral polytopes

It is already known that the automorphism group of a chiral polyhedron is never isomorphic to $PSL(2,q)$ or $PGL(2,q)$ for any prime power $q$. In this paper, we show that $PSL(2,q)$ and $PGL(2,q)$ are never automorphism groups of chiral polytopes of rank at least $5$. Moreover, we show that $PGL(2,q)$ is the automorphism group of at least one chiral polytope of rank $4$ for every $q\geq5$. Finally, we determine for which values of $q$ the group $PSL(2,q)$ is the automorphism group of a chiral polytope of rank $4$, except when $q=p^d\equiv3\pmod{4}$ where $d>1$ is not a prime power, in which c…

Point-primitive, line-transitive generalised quadrangles of holomorph type

Let $G$ be a group of collineations of a finite thick generalised quadrangle $Γ$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $Γ$, and transitively on the lines of $Γ$. We show that the primitive action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples $Γ$ for which the action of $G$ on $\mathcal{P}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where ther…

$\PSL(3,\C)$ 中的初等群

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。

Projective special linear groups $PSL_4(q)$ are determined by the set of their character degrees

Let $G$ be a finite group and let $cd(G)$ be the set of all irreducible complex character degrees of $G$. It was conjectured by Huppert in Illinois J. Math. 44 (2000) that, for every non-abelian finite simple group $H$, if $cd(G)=cd(H)$ then $G\cong H\times A$ for some abelian group $A$. In this paper, we confirm the conjecture for the family of projective special linear groups $\textrm{PSL}_4(q)$ with $q\geq 13$.

$PSL_2$ 热带化与曲面上的直线

$PSL_2$ tropicalization and lines on surfaces

本文基于Gökova会议报告，初步阐述了$PSL_2$子簇相位热带化的非阿贝尔版本，探索热带几何与二阶矩阵结合所产生的新型几何。

Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of…

Transitivity in finite general linear groups

It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme …

射影特殊线性群的全3-闭包

Total 3-closure for projective special linear groups

本文研究有限单射影特殊线性群的全3-闭包性质，明确了PSL₂(q)、PSL₃(q)、PSL₄(q)及n≥5且q>2时PSLₙ(q)的判定条件，还肯定回答了《Kourovka Notebook》中的问题20.2。

The classification of (3/2)-transitive permutation groups and (1/2)-transitive linear groups

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear groups. As a consequence we complete the determination of the finite (3/2)-transitive permutation groups -- the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the finite (k+1/2)-transitive permutation groups for integers k > 1.

二维有限射影特殊线性群共轭类的子rack与二阶同调

Subracks and second homology of the conjugacy classes of finite projective special linear groups of degree two

基于Dickson子群定理描述PSL(2,q)共轭类子rack，确定其极小非交换子rack，给出完全群共轭类关联群与Schur乘子关系的一般结果，进而明确q>3时PSL(2,q)共轭类关联群及二阶同调。

由素数阶 $2$ 和 $p$ 的元素生成 $\psl{2}{q}$

Generating $\psl{2}{q}$ by elements of prime orders $2$ and $p$

对于素数 $p\geq 5$，确定哪些 $q$ 使得有限域上的射影特殊线性群 $\psl{2}{q}$ 是 $(2,p)$-生成的，即存在阶分别为 $2$ 和 $p$ 的两个元素生成 $\psl{2}{q}$。

Permutations Which Make Transitive Groups Primitive

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and…

度为3p的非拟本原群的Erdős-Ko-Rado定理

The Erdős-Ko-Rado Theorem for non-quasiprimitive groups of degree $3p$

本文证明度为3p的非拟本原传递群的交密度必属于{1,3/2,3}，除非p为费马素数且诱导作用为包含PSL₂(q)的几乎单群。

Non-split linear sharply $2$-transitive groups

We give examples of countable linear groups in $SL_{n}(R)$ for $n \ge 3$, with no nontrivial normal abelian subgroups, that admit a faithful sharply 2-transitive action on a set. Without the linearity assumption, such groups were recently constructed by Rips, Segev, and Tent. Our examples are of permutational characteristic $2$, in the sense that involutions do not fix a point in the $2$-transitive action.

Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon $\cS$, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type, that is, the socle of $G$ is a simple Chevalley group.

Linear spaces with significant characteristic prime

Let $G$ be a group with socle a simple group of Lie type defined over the finite field with $q$ elements where $q$ is a power of the prime $p$. Suppose that $G$ acts transitively upon the lines of a linear space $\mathcal{S}$. We show that if $p$ is {\it significant} then $G$ acts flag-transitively on $\mathcal{S}$ and all examples are known.

Finite groups of units and their composition factors in the integral group rings of the groups $\text{PSL}(2,q)$

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 $G$.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSL(n,q)

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses.

Doubly transitive lines II: Almost simple symmetries

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe involving Schur covers to recover doubly transitive lines from their automorphism group. Combining our results with recent work on the affine case by Dempwolff and Kantor, we deduce a classification of all linearly dependent doubly transitive lines in real or complex space.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL(n,q)

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n,q), any finite-dimensional pointed Hopf algebra H with G(H) = PSL(n,q) or SL(n,q) is isomorphic to a group algebra.

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。

Real and strongly real classes in finite linear groups

We classify the real and strongly real conjugacy classes in $GL_n(q)$, $SL_n(q)$, $PGL_n(q)$, $PSL_n(q)$, and all quasi-simple covers of $PSL_n(q)$. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes.

有限李型单群上的有限维pointed Hopf代数 VII. PSL(n,q)和PSp(2n,q)中的半单类

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q)

该文研究有限李型单群上的有限维pointed Hopf代数分类，通过分析半单轨道上Nichols代数的维数性质，证明几类PSL、PSp群对应的有限维pointed Hopf代数均同构于群代数，完善了此前的分类工作。

Sharply 2-transitive linear groups

A group G is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left) such that G is isomorphic to the semidirect product of the multiplicative and additive groups of N. This is well known in the finite case. We prove this conjecture when G < GL(n,F) is a linear group. Here we have to assume that both the characteristic of the field F and the permutational characteristic of the group G (see Definition …

The simple classical groups of dimension less than 6 which are (2,3)-generated

In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.

Embedding central extensions of simple linear groups into wreath products

We find a criterion for the embedding of a nonsplit central extension of $PSL_n(q)$ with kernel of prime order into the permutation wreath product that corresponds to the action on the projective space.