This paper is concerned with two mani problems:(a) the determination of the conjugacy classes in the finite-dimensional unitary, symplectic and orthogonal groups over division rings or fields;(b) the determination of the equivalence classes of non-degenerate sesquilinear forms on finite-dimensional vector spaces.

Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.

Papers I, II of this projected series lay the algebraic foundations of the theory of the Clifford groups; I deals with the casep> 2, II with the casep= 2. The present introduction refers to both papers. Our theory has applications in group theory, geometry and number theory.

The present paper deals with the Clifford groups in the case p = 2. For the most part, it runs parallel to the previous paper I ([1]) on the case p > 2, and a number of proofs are therefore either given in outline or omitted. A general introduction to both papers is given in I, § 1.

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