Defining Relations in Orthogonal Groups of Characteristic Two

Abstract

Introduction. It is well-known that a group is uniquely determined by a system of generators, and a set of defining relations on those generators. Clearly it is of interest to find relations that are as simple as possible. In this paper, this question is dealt with for certain orthogonal groups of characteristic 2, which are generated by involutions.Let V be a vector space over a field K of characteristic 2 (we always exclude the prime field K = G F (2)). Let Q be a quadratic form over V, and let S be the set of orthogonal transformations of (V, Q) whose path is 1-dimensional and not contained in the radical of V. Letting O* be the group generated by S, we shall show that every relation among generators in S is a consequence of relations of length 2, 3, or 4.