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. 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 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 at most n.

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