On projective planes of type (4, m)

If a group G acts on a finite projective plane to make it a plane of type (4, m) and if G/K is the related 2-transitive representation of G then either G/K has a normal regular subgroup or PSL(2, q)G/KPL(2, q) for some prime power q.     

Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.      

On finite groups acting on ℤ<Subscript>2</Subscript>-homology 3-spheres

We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of ₂-homology 3-spheres (i.e., with the ₂-homology of the 3-sphere where ₂ denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on ₂-homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group or the binary dodecahedral group most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S³. Roughly, in the case of ₂-homology 3-spheres the groups PSL(2,5) and SL(2,5) get replaced by the groups PSL(2,q) and SL(2,q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2,q) and SL(2,q) on ₂-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on ₂-homology 3-spheres remains still open).     

Semiregular Large Sets

We develop the notion of t-homogeneous, G-semiregular large sets of t-designs, show that there are infinitely many 3-homogeneous PSL(2, q)-semiregular large sets when q 3 mod 4, two sporadic 3-homogeneous AL(1,32)-semiregular large sets, and no other interesting t-homogeneous G-semiregular large sets for t 3.     

A characterization of hadamard designs with SL(2,q) acting transitively

Given a hyperoval in a projective plane of even orderq, we can associate a Hadamard 2-design. In the case when is the Desarguesian plane P_2,q ,q=2 ^h ,h>1 and is a regular hyperoval (conic and its nucleus) then a design (q) is obtained. (q) has a point transitive automorphism group isomorphic to PSL(2,q)( SL(2,q)). We classify the designs (q) and P_2h–1,2 (the projective space of dimension 2h–1 overF ₂) among all the designsH with the same parameters as (q) admitting an automorphism groupGSL(2,q) acting transitively the points ofH. We also describe how all such designsH may be constructed and discuss the problem of when two such designs are isomorphic.This research was supported by Science and Engineering Research Council Grant GR/G 03359.     

A new characterization for some extensions of PSL(2, <Emphasis Type="Italic">q)</Emphasis> for some <Emphasis Type="Italic">q</Emphasis> by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p²) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p²) of order 2|PSL(2,p²)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p³. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.     

3‐Spontaneous Emission Error Designs from PSL(2, q) or PGL(2, q)

Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3‐SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3‐ SEEDs are derived for prime powers q and all .     

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.     

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).     

A new characterization for some linear groups

Let G be a group and π _e (G) be the set of element orders of G. Let k ? p_e(G){k\in\pi_e(G)} and m _k be the number of elements of order k in G. Let nse(G) = {m_k|k ? p_e(G)}{{\rm nse}(G) = \{m_k|k\in\pi_e(G)\}} . In Shen et al. (Monatsh Math, 2009), the authors proved that A₄ @ PSL(2, 3), A₅ @ PSL(2, 4) @ PSL(2,5){A_4\cong {\rm PSL}(2, 3), A_5\cong \rm{PSL}(2, 4)\cong \rm{PSL}(2,5)} and A₆ @ PSL(2,9){A_6\cong \rm{PSL}(2,9)} are uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse(PSL(2, q)), where q ? {7,8,11,13}{q\in\{7,8,11,13\}} , then G @ PSL(2,q){G\cong {PSL}(2,q)} .     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24     

On the Admissibility of Some Linear and Projective Groups in Odd Characteristic

A bijective mapping defined on a finite group G is complete if the mapping defined by , , is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL and PGL(2,q),q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n,q GL , admit a complete mapping.     

Commutator Subgroups of the Extended Hecke Groups \bar H(\lambda _q )

Hecke groups H(_q) are the discrete subgroups of generated by S(z) = –(z+ _q)^–1and T(z) = –1/z. The commutator subgroup of H(_q), denoted by H(_q), is studied in [2]. It was shown that H(_q) is a free group of rank q– 1.Here the extended Hecke groups obtained by adjoining to the generators of H(_q) are considered. The commutator subgroup of is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H(_q) case, the index of H(_q) is changed by q, in the case of this number is either 4 for qodd or 8 for qeven.     

Linear independence in finite spaces

The maximum number m ₂(n, q) of points in PG(n, q), n2, such that no three are collinear is known precisely for (n, q)=(n,2), (2,q), (3,q), (4, 3), (5,3). In this paper an improved upper bound of order q ^n–1 –1/2q ^n–2 is obtained for q even when n4 and q>2. A necessary preliminary is an improved upper bound for m₂(3, q), the maximum size of a k-cap not contained in an ovoid. It is shown that and that m₂(3, 4)=14.     

GBRD's: Some new constructions for difference matrices,generalised hadamard matrices and balanced generalised weighing matrices

A new construction is given for difference matrices. The generalized Hadamard matrices GH(q(q – 1)²; EA(q)) are constructed whenq andq – 1 are both prime powers. Other generalised Hadamard matrices are also shown to exist. For example, there exist GH(n; G) forn = 5² 2 3, 2⁶ 3², 11² 2² 3, 17² 2 3², 53² 2 3³, 71² 2² 3², 107² 2² 3³, 149² 5² 2 3,.... Finally, a new construction for the BGW ((q ⁴ – 1)/(q – 1),q ³,q ²(q – 1);q _q-1), and a construction for the new BGW ((q ⁸ – 1)/(q ² – 1),q ⁶,q ⁴(q ² – 1);G) are given, wheneverq is a prime power, andG is a group of orderq + 1.     

Twisted honeycombs {3, 5, 3} t and their groups

A honeycomb of type {3, 5, 3} is finite when it has Petrie polygons of length 4, 6, 7 or 9. In these cases its group of automorphisms is isomorphic to PSL(2, 9), PSL(2, 11)×₂, PSL(2, 29) or PSL(2, 19). Its edges and faces are incident in a manner indicated by the vertices of a bipartite graph of girth 8 (in the first case) or 10, whose group is PGL(2, 9), PGL(2, 11)×₂, PSL(2, 29)×₂ or PGL(2, 19), respectively.     

Characterization of {(q + 1) + 2, 1;t,q}-min · hypers and {2(q + 1) + 2, 2; 2,q}-min · hypers in a Finite projective geometry

In this paper, we shall characterize all {(q + 1) + 2, 1;t, q}-min · hypers and all {2(q + 1) + 2, 2; 2,q}-min · hypers for any integert 2 and any prime powerq 3. In the next paper [8], we shall characterize all {2(q + 1) + 2, 2;t, q}-min · hypers for any integert 3 and any prime powerq 5 using the results in this paper.     

Spreads of T 2(o), α-flocks and Ovals

We establish a representation of a spread of the generalized quadrangle T ₂(0), o an oval of PG(2, q), q even, in terms of a certain family of q ovals of PG(2, q) and investigate the properties of this representation. Using this representation we show that to every flock of a translation oval cone in PG(3, q) (-flock), q even, there corresponds a spread of T ₂(o) for an oval o determined by the -flock. This gives constructions of new spreads of T ₂(o), for certain ovals o, and in some cases solves the existence problem for spreads. It also provides a geometrical characterization of the ovals associated with a flock of a quadratic cone.     

Triads, Flocks of Conics and Q -(5,q)

We show that if an ovoid of Q (4,q),q even, admits a flock of conics then that flock must be linear. It follows that an ovoid of PG (3,q),q even, which admits a flock of conics must be an elliptic quadric. This latter result is used to give a characterisation of the classical example Q ^-(5,q) among the generalized quadrangles T ₃( ), where is an ovoid of PG (3q) and q is even, in terms of the geometric configuration of the centres of certain triads.     

On a subalgebra of the centre of a group ring II

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Külshammer generalized this result to blocks with abelian defect groups. Here, we show that is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore and are algebras for all primes p and all prime powers q. Furthermore we prove that is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|. Received: 18 May 2007