A proof by graphs that PSL(2, 7) PSL(3, 2)

We give three definitions of the Coxeter graph. By the second one we see that PSL(2, 7) is contained in the automorphism group of that graph as a subgroup of index 2, and by the third one that the same holds for PSL(3, 2).     

On PSL(2,q) as a totally irregular collineation group

Non-abelian simple totally irregular collineation groups containing an involutorial perspectivity have been classified by the authors in a recent paper. They are PSL(2,q), PSL(3,q), PSU(3,q), Sz(q), the alternating group on 7 letters, and the second Janko sporadic simple group. In this article, we study PSL(2,q),q congruent to 1 modulo 4, as a collineation group containing an involutory homology.C. Y. Ho was partially supported by a NSA grant.     

Halving PSL(2,q)

We show that PSL(2,q), q 3(mod 4), contains a subset of half the cardinality of PSL(2,q), which is uniformly 2-homogeneous on the projective line.     

Coprime commutators in PSL(2, q)

We show that every element of PSL(2, q) is a commutator of elements of coprime orders. This is proved by showing first that in PSL(2, q) any two involutions are conjugate by an element of odd order.     

On the genus of the groups PSL(2, q), PSL(3, q), and PSp(4, q)

SL(n, q) is the group of n×n matrices, over the Galois field GF(q), of determinate 1. PSL(n, q) is SL(n, q) modulo the scalar n×n matrices of determinate 1. PSL(n, q) acts on the Desarguesian projective space PG(n−1, q). Sp(4, q) is the group of 4 × 4 matrices of determinate 1 which preserve the symplectic bilinear form on the 4 × 1 matrices over GF(q). PSp(4, q) is Sp(4, q) modulo Z = {1,−1}. PSp(4, q) acts on the symplectic generalized quadrangle W(3, q), a subspace of the projective space PG(3, q), as a group of automorphisms. In this paper, bounds are given for the genus of these groups.     

Weyl chamber flow on irreducible quotients of PSL(2, ℝ) × PSL(2, ℝ)

We study the topological dynamics of the action of the diagonal subgroup on quotients Γ\PSL(2, ℝ) × PSL(2, ℝ), where Γ is an irreducible lattice. Closed orbits are described and a set of points of dense orbit is explicitly given. Such properties are expressed using the Furstenberg boundary of the associated symmetric space ℍ × ℍ.     

Writing Elements of PSL(2, q) as Commutators

We prove that for q ≥ 13, an element A of SL(2, q) is the commutator of a generating pair if and only if A ≠ ?I and the trace of A is not 2. Consequently, when q is odd and q ≥ 13, every nontrivial element of PSL(2, q) is the commutator of a generating pair, and when q is even, an element of PSL(2, q) is the commutator of a generating pair if and only if its trace is not 0. The proof of these results also leads to an improved lower bound on the number of T-systems of generating pairs of PSL(2, q).     

Geometries of the Group PSL(2,11)

We determine all residually weakly primitive flag-transitive geometries for the groups PSL(2,11) and PGL(2,11). For the first of these we prove the existence by simple constructions while uniqueness, namely the fact that the lists are complete, relies on MAGMA programs. A central role is played by the subgroups Alt(5) in PSL(2,11). The highest rank of a geometry in our lists is four. Our work is related to various atlases of coset geometries.     

3‐Designs of PSL(2, 2n) With block sizes 4 and 5

We determine the distribution of 3‐designs among the orbits of 4‐ and 5‐element subsets under the action of PSL(2,2ⁿ) on the projective line. Thus we give complete information on all Kramer–Mesner matrices for the action of PSL(2,2ⁿ) on 3‐sets versus 4‐ and 5‐sets. As a consequence, all 3‐designs with block sizes 4 and 5 and automorphism group PSL(2,2ⁿ) can immediately be obtained. © 2003 Wiley Periodicals, Inc.     

2-(v,k,1)设计和PSL(3,q)(q是奇数)

§ 1　 IntroductionA2 -(v,k,1 ) design D=(S,B) consists ofa finite set Sof v points and a collection Bof some subsets of S,called blocks,such that any two points lie on exactly one blockand each block contains exactly k points.A flag of Dis a pair(α,B) such thatα∈S,B∈Bandα∈B,the set of all flags is denoted by F.We assume that2≤k≤v.An automorphism of Dis a permutation of the points which leaves the set Binvari-ant,all the automorphisms form a group Aut D.Let G be a subgroup of A…     

Flag-Transitive 3-(v,k, 3) Designs and PSL(2, q) Groups

This article is a contribution to the study of the automorphism groups of 3-(v,k,3) designs.Let S =(P,B) be a non-trivial 3-(q+ 1,k,3) design.If a two-dimensional projective linear group PSL(2,q) acts flag-transitively on S,then S is a 3-(q + 1,4,3) or 3-(q + 1,5,3) design.     

Arc-transitive cubic Cayley graphs on PSL(2,p)

For a prime p at least 5,let T=PSL(2,p).This paper gives a classification of the connected arc-transitive cubic Cayley graphs on T and a determination of the gener- ated pairs ((?),(?)) of T such that o((?))=2 and o((?))=3.     

Simple 3‐designs of PSL(2, 2n) with block size 7

In this paper, we determine the number of the orbits of 7‐subsets of with a fixed orbit length under the action of PSL(2, 2ⁿ). As a consequence, we determine the distribution of λ for which there exists a simple 3‐(2ⁿ + 1, 7, λ) design with PSL(2, 2ⁿ) as an automorphism group. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 1–17, 2008     

PSL（2，F）的一个嵌入定理及其应用

设Ｆ是任意域，Ｇ代表ＳＬ（２，Ｆ）或ＰＳＬ（２，Ｆ）．本文的主要结果是：设Ｋ是Ｆ的子域，则Ｇ中同构于ＳＬ（２，Ｋ）或ＰＳＬ（２，Ｋ）的子群在Ｇ的自同构的作用下彼此共轭，利用这一结果，本文明确确定了A₁^[1]型的仿射Ｋａｃ－Ｍｏｏｄｙ群的一类极大正规子群．     

A note on the amenable subgroups of PSL (2,R)

In this note we give a characterization of the amenable subgroups of PSL (2,R) in terms of the action on the hyperbolic half-plane.This work was partially supported by G.N.A.F.A. of the C.N.R., Italy.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

The classification of discrete 2-generator subgroups of PSL(2,R)

This paper gives a short geometric algorithm for deciding the discreteness of most 2-generator subgroups of PSL(2,R), as well as a self-contained algorithmic approach to the complete classification. Partially supported by NSF Grant No. MCS-7801248     

旗传递点本原且基柱为${\rm PSL}_n(q)$的$(v,k,4)$-对称设计

若$\cal D$为一个非平凡旗传递点本原对称$(v,k,4)$设计, 其基柱为${\rm PSL}_n(q)$且$G\leq {\rm Aut}(\cal D)$. 那么, $\cal D$ 必为$2$-$(15,8,4)$设计且${\rm Soc}(G)={\rm PSL}_2(9)$.     

Division Algebras with PSL(2, q)-Galois Maximal Subfields

If G is a finite group and k is a field, then G is k-admissible if there exists a G-Galois extension L/k such that L is a maximal subfield of a k-division algebra. We prove that PSL(2, 7) is k-admissible for any number field which either fails to contain or which has two primes lying over the dyadic prime. In addition, PSL(2, 11) is shown to be admissible over or any number field k with at least two extensions of the dyadic prime. Indeed, there exist infinitely many linearly disjoint admissible extensions for these groups.