The Elements of the Orthogonal Group Ω (V) as Products of Commutators of Symmetries

Abstract

We will assume throughout that F is a field of characteristic char F / 2 and that V is a non-degenerate quadratic space over F of finite dimension Ž . Ž . dim V s n. The orthogonal group of V is denoted O V and ⍀ V is its n n commutator subgroup. John Hsia, in reference to the classical fact that Ž . every element of ⍀ V is a product of commutators of symmetries, asked n

Involutions and commutators in orthogonal groups

“…As B°°(7r) = V we obtain from Lemma 4.3 that 0(TT) = det(l -n)dV = ^(l)d(V) = (2 + a)dV, and 0 ( -J T ) = q(-\)d(V) = (2 -a)dV. Hence, dV = 0 ( -l v ) = @(-iut) = (4-a 2 )K* 2 ^ -K* 2 . This implies that V is not a hyperbolic plane.…”

“…The following proposition was proved in [10]. (a) If it-type( V) = 2* or 3 then dV = p(l)p(-l)K* 2 and @(n) = p(-l)K* 2 .…”

“…PROOF. We have q := mip(7r) = x 2 + ax + 1 for some a € K. As mip(7r) is irreducible it follows that a 2 -4 £ K* 2 . The characteristic polynomials of 1 -n, respectively 1 + n, are ^(1 -x) € K[x], respectively q(x -1).…”

“…Now let ^ be the symmetric bilinear form whose Gram matrix (g, ; ) is given by gij '•= f.j when / < j and g tJ := f }J when y < /. Then ind(V, g) = (n -l)/2 = ind(V, / ) and d(V, g) = XK* 2…”

Applications of the wall form to unipotent isometries of index two

“…The Wall form was applied in [5] and [7] in connection with the Zassenhaus decomposition [19] of isometries in characteristic = 2. It was also used in [6] to study the length of elements in the commutator subgroup of the orthogonal group. In characteristic 2, this form was applied in [1] to study the relation between the commutator subgroup of the orthogonal group and the spinorial kernel of a nondegenerate defective quadratic space.…”