THE z-CLASSES OF ISOMETRIES

Let G be a group. Two elements x, y are said to be in the same z-class if their centralizers are conjugate in G. The conjugacy classes give a partition of G. Further decomposition of the conjugacy classes into z-classes provides an important information about the internal structure of the group. Let V be a vector space of dimension n over a field F of characteristic different from 2. Let B be a non-degenerate symmetric, or skew-symmetric, bilinear form on V. Let I(V, B) denote the group of isometries of (V, B). We parametrize the z-classes in I(V, B). We show that the number of z-classes in I(V, B) is finite when F is perfect and has the property that it has only finitely many field extensions of degree ≤ n. Along the way we also determine the conjugacy classes in I(V, B).