Caps in the projective geometry PG(n,4)

We construct a cap C_n in PG(n,4) of size 12(2ⁿ⁻²−1)+5. In the cases when n=3 and 4, this yields geometric constructions of the well-known ovoid in PG(3,4) and the Tallini cap in PG(4,4).     

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.     

The Haemers partial geometry and the Steiner system S(5, 8, 24)

Haemers has constructed a partial geometry with parameters s = 4, t = 17, and α = 2, using properties of the Hoffman-Singleton graph. We describe this geometry in terms of the Steiner system S(5, 8, 24).     

Sets of type (m,n) in the affine and projective planes of order nine

In this paper we give the results of exhaustive computer searches for sets of points of type (m, n) in the projective and affine planes of order nine. In particular, as the list of planes of order nine is known to be complete, our results are also complete. We also examine all known constructions of sets of type (m, n) that apply to the planes of order nine, in an attempt to summarise and extend all existing knowledge about such sets. The contrast between known constructions and our computer results leads us to conclude that sets of type (m, n) are far more numerous than was previously thought.     

Sets of type (0,n) in Steiner systems and in finite projective planes

In this paper we study the sets of type (0,n)₁ in a finite projective plane and the inclusion relations between them.     

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

A characterization of 3-local geometry of M(24)

The largest Fischer 3-transposition group M(24) acts flag-transitively on a 3-local incidence geometry (M(24)) which is a c-extension of the dual polar space associated with the group O ₇(3). The action of the simple commutator subgroup M(24) is still flag-transitive. We show that (M(24)) is characterized by its diagram under the flag-transitivity assumption. The result implies in particular that (M(24)) is simply connected. The geometry (M(24)) appears as a subgeometry in the Buekenhout-Fischer 3-local geometry (F ₁) of the Monster group. The simple connectedness of (M(24)) has played a crucial role in the characterization of (F ₁), which has been achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushing-up result by U. Meierfrankenfeld.Dedicated to Professor B. Fischer on the occasion of his sixtieth birthday     

Characterization of the graphs of the Johnson schemes G(3k, k) and G(3k + 1, k)

The graphs of the Johnson schemes G(3k, k) and G(3k + 1, k) are characterized by their parameters. In particular this finishes the characterization of the tetrahedral graphs G(n, 3).     

Congruences modulo 8 for the class numbers of Q(√±p), p ≡ 3(mod 4) a prime

A congruence modulo 8 is proved relating the class numbers of the quadratic fields Q(√p) and Q(√?p), where p is a prime congruent to 3 modulo 4.     

Finite projective planes admitting a projective linear group PSL (2, q)

Let S be a projective plane, and let G?Aut(S) and PSL(2, q) ? G ? PΓL(2, q) with q > 3. If G acts point-transitively on S, then q = 7 and S is of order 2.     

Some resolutions of S(5, 8, 24)

There exist 13 mutually disjoint resolutions of the Steiner system S(5, 8, 24). There also exist nine nonisomorphic mutually disjoint resolutions of S(5, 8, 24) where three of the resolutions have the same L₂(23) as an automorphism group and the other six have the same affine group C₂₃¹¹ as an automorphism group. A resolution of S(5, 8, 24) using a group of order 21 is displayed and a 13-dimensional Room-type design is mentioned.     

On projective planes of type (5, m)

If a group acts on a finite projective plane to make it a plane of type (5,m) then the related 2-transitive permutation representation of it has a normal regular subgroup.Some of the research which is reported in this paper was carried out at the Universität Kaiserslautern while I was a von Humboldt Fellow. I wish to thank the von Humboldt Stiftung for their support while I was in Germany.