On element-centralizers in finite groups

For any group G, let |Cent(G)| denote the number of centralizers of its elements. A group G is called n-centralizer if |Cent(G)| = n. In this paper, we find |Cent(G)| for all minimal simple groups. Using these results we prove that there exist finite simple groups G and H with the property that |Cent(G)| = |Cent(H)| but ${G\not\cong H}$ . This result gives a negative answer to a question raised by A. Ashrafi and B. Taeri. We also characterize all finite semi-simple groups G with |Cent(G)| ≤  73.