ABSTRACT. A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ(G). Determining σ(G) is an open problem for many non-solvable groups. For symmetric groups S n , Maróti determined σ(S n ) for odd n with the exception of n = 9 and gave estimates for n even. In this paper we determine σ(S n ) for n = 8, 9, 10 and 12. In addition we find the covering number for the Mathieu group M 12 and improve an estimate given by Holmes for the Janko group J 1 .

The only known skew-translation generalised quadrangles (STGQ) having order (q, q), with q even, are translation generalised quadrangles. Equivalently, the only known groups G of order q 3 , q even, admitting an Ahrens-Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new structural information for a group G admitting an AS-configuration, (ii) a classification of the STGQ of order (8,8), and (iii) a classification of the STGQ of order (q, q) for odd q (using work of Ghinelli and Yoshiara).Crown

The covering number of a group G, denoted by σ(G), is the size of a minimal collection of proper subgroups of G whose union is G. We investigate which integers are covering numbers of groups. We determine which integers 129 or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most 129 by introducing effective new computational techniques. Furthermore, we prove that, if F 1 is the family of finite groups G such that all proper quotients of G are solvable, then N − {σ(G) : G ∈ F 1 } is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form (q m − 1)/(q − 1), where m = 3 and q is a prime power, is a covering number, generalizing a result of Cohn.2010 Mathematics Subject Classification. Primary 20D60; Secondary 20B15.

If a group G is the union of proper subgroups H 1 , . . . , H k , we say that the collection {H 1 , . . . H k } is a cover of G, and the size of a minimal cover (supposing one exists) is the covering number of G, denoted σ(G). Maróti showed that σ(S n ) = 2 n−1 for n odd and sufficiently large, and he also gave asymptotic bounds for n even. In this paper, we determine the exact value of σ(S n ) when n is divisible by 6.

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P . We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group G acting regularly on the point set of a generalized quadrangle of order (u 2 , u 3 ) or (s, s), where s is odd and s + 1 is coprime to 3, cannot have any nonabelian minimal normal subgroups.2010 Mathematics Subject Classification. Primary 51E12, 05B25.

Ostrom and Wagner (1959) proved that if the automorphism group G of a finite projective plane π acts 2-transitively on the points of π, then π is isomorphic to the Desarguesian projective plane and G is isomorphic to PΓL(3, q) (for some prime-power q). In the more general case of a finite rank 2 irreducible spherical building, also known as a generalized polygon, the theorem of Fong and Seitz (1973) gave a classification of the Moufang examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group G acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to G being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.2010 Mathematics Subject Classification. Primary 51E12, 20B05, 20B15, 20B25. 1 2 JOHN BAMBERG, CAI HENG LI, AND ERIC SWARTZ Conjecture 1.1 (W. M. Kantor 1991). If Q is a finite flag-transitive generalized quadrangle and Q is not a classical generalized quadrangle, then (up to duality) Q is the unique generalized quadrangle of order (3, 5) or the generalized quadrangle of order (15, 17) arising from the Lunelli-Sce hyperoval.Finite generalized polygons satisfying stronger symmetry assumptions, such as the Moufang condition [15] or distance-transitivity [12,28], have been classified. The current state-of-the-art for generalized quadrangles is the classification of antiflag-transitive finite generalized quadrangles in [5], where it was shown that, up to duality, the only nonclassical antiflag-transitive generalized quadrangle is the unique generalized quadrangle of order (3,5). (An antiflag is a non-incident point-line pair.) Notice that by the GQ Axiom, antiflag-transitivity implies flag-transitivity for a generalized quadrangle.The aim of this paper is to provide further progress toward Conjecture 1.1. We study finite generalized quadrangles with a strictly weaker local symmetry condition, local 2transitivity. If G is a subgroup of collineations of a finite generalized quadrangle Q that is transitive both on pairs of collinear points and pairs of concurrent lines, then Q is said to be a locally (G, 2)-transitive generalized quadrangle.Our main result is a complete classification of thick locally 2-transitive generalized quadrangles, the proof of which relies on the Classification of Finite Simple Groups (CFSG) [20].Theorem 1.2. If Q is a thick finite locally (G, 2)-transitive generalized quadrangle and Q is not a classical generalized quadrangle, then (up to duality) Q is the unique generalized quadrangle of order (3, 5).An equivalent definition of a locally 2-transitive generalized quadrangle is that it has an incidence graph that is locally 2-arc-transitive; see Section 2 for details. Since an equivalent definition of an antiflag-transitive generalized quadrangl...

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