On element-centralizers in finite groups

摘要

For any group G, let vertical bar Cent(G)vertical bar denote the number of centralizers of its elements. A group G is called n-centralizer if vertical bar Cent(G)vertical bar = n. In this paper, we find vertical bar Cent(G)vertical bar for all minimal simple groups. Using these results we prove that there exist finite simple groups G and H with the property that vertical bar Cent(G)vertical bar = vertical bar Cent(H)vertical bar but G not similar or equal to 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 vertical bar Cent(G)vertical bar <= 73.