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On element-centralizers in finite groups

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Abstract

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.

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  1. Department of Mathematics, University of Kurdistan, P.O. Box 416, Sanandaj, Iran

    Mohammad Zarrin

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  1. Mohammad Zarrin
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Correspondence to Mohammad Zarrin.

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Zarrin, M. On element-centralizers in finite groups. Arch. Math. 93, 497–503 (2009). https://doi.org/10.1007/s00013-009-0060-1

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