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Let G be a group. Two elements x,y ∈ G are said to be in the same z-class if their centralizers in G are conjugate within G. Consider F a perfect field of characteristic ≠ 2, which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field F_0 has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of z-classes in the unitary group over such fields is finite. Further, we count the number ofarXiv:1610.06728v2 fatcat:kic2woswbrfdnki5qf3cdusboq