The z-classes of isometries

Abstract

LetGbe a group. Two elementsx, yare said to be in thesamez-classif their centralizers are conjugate inG. LetVbe a vectorspace of dimensionnover a fieldFof characteristic different from 2.LetBbe a non-degenerate symmetric, or skew-symmetric, bilinearform onV. Let I(V, B) denote the group of isometries of (V, B). Weshow that the number ofz-classes in I(V, B) is finite whenFis perfectand has the property that it has only finitely many field extensions ofdegree at mostn