On the structure of unitary groups

1. Let K be an arbitrary sfield with an involution J, that is, a one-to-one mapping £—*f of K onto itself, distinct from the identity, such that (£ + rj)J = ¡-J-\-r]J, (¡-riY—i]J¿,J, and (¿/)/=?. Let E be an «-dimensional right vector space over K (n ^ 2) ; an hermitian (resp. skew-hermitian) form over £ is a mapping (x, y)—*f(x, y) of EXE into iC which, for any x, is linear in y, and such that/(y, x) = (f(x, y))J (resp./(y, x) = — (/(x, y))J). This implies that /(x, y) is additive in x and such that/(xA, y) =X//(x, y). The values/(x, x) are always symmetric (resp. skew-symmetric) elements of K, that is, elements a such that a/=o; (resp. aJ = —a). The orthogonality relation/(x, y) =0 relative to / is always symmetric. We shall always suppose that the form/is nondegenerate, or in other words that there is no vector in E other than 0 orthogonal to the whole space. Moreover, when the characteristic of K is 2, the distinction between hermitian and skew-hermitian forms disappears, and /(x, x) is symmetric for every xG-E; in that case we shall make the additional assumption that/(x, x) has always the form ¿--f-f ("trace" of £) for a convenient ¿£i?; this assumption is automatically verified when the restriction of / to the center Z of K is not the identity, but not necessarily in the other cases. A unitary transformation « of £ is a one-to-one linear mapping of E onto itself such that/(«(x), u{y)) =/(x, y) identically; these transformations constitute the unitary group U„(K, f). In a previous paper [5, pp. 63-82](1), I have studied the structure of that group in the two simplest cases, namely those in which K is commutative, or if is a reflexive sfield and the form / is hermitian; the present paper is devoted to the study of Un(K, f) in the general case. 2. We shall need the following lemma: