# On finite groups with a certain number of centralizers

Article

## Abstract

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n.

In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)≃A5, then #Cent(G)=22 or 32. Moroever, we prove that A5 is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A5 in terms of the number of centralizers

20D99 20E07

### Key Words and phrases

Finite group n-centralizer group primitiven-centralizer group simple group

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