A new efficient presentation for PSL(2,5) and the structure of the groups G(3,m,n)

G(3, m, n) is the group presented by . In this paper, we study the structure of G(3, m, n). We also give a new efficient presentation for the Projective Special Linear group PSL(2, 5) and in particular we prove that PSL(2, 5) is isomorphic to G(3, m, n) under certain conditions.     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

The structure of symmetry groups of GK(2n,G)

The automorphism groups of the one-factorizations GK(2n,G) are computed. It is shown that every 1-factorization of K_2n with a subgroup of the automorphism group that acts sharply 2-transitively on the one-factors must be GK(p^m + 1, (Z_p)^m) for some odd prime p. © 1994 John Wiley & Sons, Inc.     

Notes on Lie Algebra \Sigma(n, m, r, G)

Supplementary discussions are given to the Lie algebra \Sigma(n, m, r, G). Minor errors in some formulas of a previous paper (see Chin. Ann. of Math., 4B(3), 1983, 329—346) are corrected.     

Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both G( n,p) and G( n,m), with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combine probability space, of the events $\{a\to s\}$ and $\{s\to b\}$ . For G(n,p), we prove that there is a $pc = 1/2$ such that for a fixed $p < pc$ the correlation is negative for large enough n and for $p > pc$ the correlation is positive for large enough n. We conjecture that for a fixed $n \ge 27$ the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/({{n}\atop {2}})$ , there is a critical pc that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute \input amssym $\Bbb{P}(a\to s)$ and \input amssym $\Bbb{P}(a\to s, s\to b)$ . We also briefly discuss the corresponding question in the quenched version of the problem. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

New 5-designs with automorphism group PSL(2, 23)

Blocks of the unique Steiner system S(5, 8, 24) are called octads. The group PSL(2, 23) acts as an automorphism group of this Steiner system, permuting octads transitively. Inspired by the discovery of a 5-(24, 10, 36) design by Gulliver and Harada, we enumerate all 4- and 5-designs whose set of blocks are union of PSL(2, 23)-orbits on 10-subsets containing an octad. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 147–155, 1999     

Characterization of the linear groups PSL(2,11) and PSL (2,13)

We prove the nonsimplicity of a finite group containing an involution τ such that the quotient group C(τ)/{τ} the Frobenius group with an additional factor of odd prime order acting transitively on the nonunit elements of the kernel. Based on this we obtain a characterization of the linear groups PSL(2,11) and PSL(2,13).     

An inductive approach to representations of complex reflection groups G(m, 1, n)

We propose an inductive approach to the representation theory of the chain of complex reflection groups G(m, 1, n). We obtain the Jucys-Murphy elements of G(m, 1, n) from the Jucys-Murphy elements of the cyclotomic Hecke algebra and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. We construct representations of G(m, 1, n) using a new associative algebra whose underlying vector space is the tensor product of the group ring ?G(m, 1, n) with a free associative algebra generated by the standard m-tableaux.     

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.     

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.     

On the Action of the Groups GL(n+1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) on PG(n, q t)

Let q be a prime power and let n ≥ 0, t ≥ 1 be integers. We determine the sizes of the point orbits of each of the groups GL(n + 1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) acting on PG(n, q ^t) and for each of these sizes (and groups) we determine the exact number of point orbits of this size.     

Schottky groups can not act on P C 2n as subgroups of PSL 2n+1(C)

In this paper we look at a special type of discrete subgroups of PSL _n+1(C) called Schottky groups. We develop some basic properties of these groups and their limit set when n > 1, and we prove that Schottky groups only occur in odd dimensions, i.e., they cannot be realized as subgroups of PSL _2n+1(C).