For any group G, let C(G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n. In this note, we find |C(G)| for all minimal simple groups. Using these results we prove that there exist finite simple groups G and H with the property that |C(G)| = |C(H)| but G ∼ = H. This result gives a negative answer to a question raised by A. Ashrafi and B. Taeri. Finally we characterize all n-centralizer finite semi-simple groups for n ≤ 73.
