All the overgroups of SU(n, K, f) or I2(n, K, Q) in GL(n, K) are determined, for arbitrary division rings K and arbitrary forms for Q with positive Witt indices.
Let K be a division ring with an involutory anti-automorphism J: a ~ ~, and let V be an n-dimensional left space over K. A J-sesquilinear form on F means a biadditive mapping f: V x V~ K satisfying f(ax, by) = af(x, y)/Tfor all x, y ~ V and a, b ~ K. Let e = __+ 1 E K*. An e-Hermitian form on V means a J-sesquilinear form f satisfying f(x, y) = ef(y, x) for all x, y e V. Such an f is said to be trace-valued if allf(x, x) e {a + e~ I a e K }. A non-degenerate tracevalued c-Hermitian form f determines a unitary group U(n,K,f)= {g~GL(V) lf(xg, yg ) =f(x,y) for all x,y~ V}. In the special case J = 1, U(n, K,f) is a symplectic group Sp(n, K,f) (when ~ = -1) or an orthogonal group O(n, K,f) (when char K ~ 2 and ~ = 1). When K is a field and J = 1, we can define a quadratic form Q on v, and consider the orthogonal group O(n,K, Q) = {gEGL(V) IQ(x9) = Q(x) for all x~ V} and its commutator group f~(n, K, Q). Each such Q associates with a symmetric form f defined by f(x,y)= Q(x + y)-Q(x)-Q(y) for all x, ye V. We have O(n, K, Q) = O(n, K,f) when char K ~ 2, while O(n, K, Q) < Sp(n, K,f) when char K = 2. When char K = 2, a regular Q may associate with a degenerate f, in such case we can regard O(n,K,Q)
this paper, for any field F with char F = 2, any quadratic form Q on an F-space V= V(2m, F) associated with a non-degenerate symmetric formf and any F2-subspace L of F, we shall denote by O(2m, F, Q, L) the orthogonal group {g ~ Sp(2m, F, f) l Q(xo) = Q(x) (rood L)}. And we denote by f~(2m, F, Q, L) the commutator group of O(2m, F, Q, L). In particular, we have O(2m, F, Q, 0) = O(2m, F, Q) and O(2m, F, Q, F) = Sp(2m, F,f).
The purpose of this paper is to determine the overgroups in GL(n, K) of SU(n, K,f) or f2(n, K, Q) with positive Witt index v = v(f) or v(Q). We have the following results.