z -Classes in Groups Ravindra Kulkarni, Rahul Kitture, Vikas Jadhav September 4, 2013 Abstract For a group G, we say that x, y ∈ G are in the same z -class if their centralizers are conjugate in G. The vague notion of “dynamical types” of transformations in classical geometries is closely related to the precise notion of z -classes of elements in the automorphism groups of these geometries, cf. [7, 8, 12]. Also the number of z -classes, and more generally, the number of “z -classes of abelian groups”, are numerical invariants for the equivalence class of isoclinic groups, cf. [10]. In this paper, we determine the number of z -classes in some finite p-groups, and we obtain the upper and lower bounds for the number of z -classes of non-abelian p-groups. The lower bound is p +2, and we show that a necessary and sufficient condition for G to attain the lower bound is that either G/Z (G) is of order p 2 or else G has an abelian subgroup of index p and G/Z (G) is of maximal class. Also, we determine the number of z -classes in certain families of p-groups. 1 Introduction Let G be a group acting on a set X . For x ∈ X , let G x denote the stabilizer subgroup of G at x. We say that x,y ∈ X are in the same orbit class if G x and G y are conjugate in G. We shall denote this equivalence relation by x ∼ 0 y. Let R(x) denote the equivalence class of x with respect to ∼ 0 . When X = G and G acts on itself by conjugation then the stabilizer group of an element is its centralizer. In this case we call R(x), the z -class of x ∈ G. The dynamical types of transformations in classical geometries are closely related to the the z -classes of elements in the automorphism groups of these geometries. For example, let G be the group of orientation preserving isome- tries of the Euclidean plane E 2 . A classical theorem asserts that, each non- identity, orientation preserving, isometry of E 2 is either a translation or a rotation around a point of E 2 . We usually say that the translation and the 1