Common Hyperplane Medians for Random Vectors

xpm -x and necessarily splits in K[X] into distinct first degree factors so that Da is diagonalizable. Applying this to Dg, for Ek = {x: [g, x] = kx}, we have Eo K and D = YEk, where the sum is direct and taken over all k E K with Ek # 0. Now, if x E D *, gx xg kx for some k E K is equivalent to requiring that x belong to N *. Moreover, y E Ek is equivalent to y E Kx. Then each Ek is a K-subspace of dimension 1 and Ek* ig the coset K *x in N *. Hence, dimKD = q. From the structure of finite fields it follows readily that K is a Galois extension of Z. We can identify N *7K * with a subgroup of G(K/Z) and, if J is the fixed field for N */K *, a E J implies xax-1 = a for all x E N*. Then Da is zero on each Ek so that Da = 0 and a E Z. Hence N *7K * G(K/Z), implying dimzK IN*/K*l = q. Combining the results above leads to dimzD= (dimKD)(dimzK) = q2 and dimZC(b) = q for all b E D, b 5 Z. If r = jZj then