We show that an elation generalized quadrangle that has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (that is, is a flock quadrangle).

The natural geometric setting of quadrics commuting with a Hermitian surface of ${\rm PG}(3,q^2)$, q odd, is adopted and a hemisystem on the Hermitian surface ${\cal H}(3,q^2)$ admitting the group $\mathrm{{P}}\Omega^-(4,q)$ is constructed, yielding a partial quadrangle ${\rm PQ}((q-1)/2,q^2,(q-1)^2/2)$ and a strongly regular graph srg$((q^3+1)(q+1)/2,(q^2+1)(q-1)/2,(q-3)/2,(q-1)^2/2)$. For $q>3$, no partial quadrangle or strongly regular graph with these parameters was previously known, whereas when $q=3$, this is the Gewirtz graph. Thas conjectured that there are no hemisystems on ${\cal H}(3,q^2)$ for $q>3$, so these are counterexamples to his conjecture. Furthermore, a hemisystem on ${\cal H}(3,25)$ admitting $3.A_7.2$ is constructed. Finally, special sets (after Shult) and ovoids on ${\cal H}(3,q^2)$ are investigated.

New proofs are given of the fundamental results of Bader, Lunardon and Thas relating flocks of the quadratic cone in PG(3, q), q odd, and BLT-sets of Q(4, q). We also show that there is a unique BLT-set of H(3, 9). The model of Penttila for Q(4, q), q odd, is extended to Q(2m, q) to construct partial flocks of size qm/2+m/2 – 1 of the cone kin PG(2m – 1, q) with vertex a point and base Q(2m – 2, q), where q is congruent to 1 or 3 modulo 8 and m is even. These partial flocks are larger than the largest previously known for m > 2. Also, the example of O'Keefe and Thas of a partial flock of k in PG(5, 3) of size 6 is generalised to a partial flock of the cone k of PG(2pn – 1, p) of size 2pn, for any prime p congruent to 1 or 3 modulo 8, with the corresponding partial BLT-set of Q(2pn, p) admitting the symmetric group of degree 2pn + 1.

In this paper we obtain a classification of those subgroups of the finite general linear group $\mbox{GL}_d (q)$ with orders divisible by a primitive prime divisor of $q^e - 1$ for some $e > \frac{1}{2}d$. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups.

An ovoid in a 3-dimensional projective geometry PG(3, q) over the field GF(q), where q is a prime power, is a set of q2+1 points no three of which are collinear. Because of their connections with other combinatorial structures ovoids are of interest to mathematicians in a variety of fields; for from an ovoid one can construct an inversive plane [3], a generalised quadrangle [11], and if q is even, a translation plane [13]. In fact the only known finite inversive planes are those arising from ovoids in projective spaces (see [2]). Moreover, there are only two classes of ovoids known, namely the elliptic quadric and, for q even and not a square, the Tits ovoids; these will be described in the next section. If the field order q is odd, then it was shown by Barlotti and Panella (see [3, 1.4.50]) that the only ovoids are the elliptic quadrics. This paper contains a geometrical characterisation of the two known classes of ovoids.

This paper studies o-polynomials, that is, polynomials which represent hyperovals in Desarguesian projective planes of even order. We present theoretical restrictions on the form that O-polynomials can have, and we determine the number of o-polynomials corresponding to each of the known classes of hyperovals (other than Cherowitzo's). We use this to give the number of known o-polynomials for the fields of orders 4, 8, 16 and 32. Exploratory computer searches for o-polynomials for fields of small orders greater than 16 are reported.