The Suzuki groups form one of the 17 families of non-abelian finite simple
groups. They can be defined in various ways as groups of 4 × 4 matrices over
finite fields of order q = 2^{2n+1} , for n > 0.

The best description of a group is usually as the automorphism group of
some object. Most descriptions of the Suzuki groups involve a *geometrical*
object called an *ovoid*, which consists of q^{2} + 1 subspaces of dimension 1
in a 4-dimensional space, with the property that any three of them span a
3-dimensional subspace. However, constructing these ovoids usually involves
some kind of `magic' (such as pulling them out of a hat, like rabbits).

In this talk I shall describe a new family of *algebraic* objects, whose automorphism groups are the Suzuki groups, and which give rise to the Suzuki
ovoids in a natural manner. Thus we obtain a new and elementary definition
(and description) of the Suzuki groups. Very little actual group theory is
involved, just a little linear algebra.