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Paper   IPM / M / 49
School of Mathematics
  Title:   On finite groups with a given number of centralizers
  Author(s):  A. R. Ashrafi
  Status:   Published
  Journal: Algebra Colloq.
  No.:  2
  Vol.:  7
  Year:  2000
  Pages:   139-146
  Supported by:  IPM
For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G)=|Cent(G)|. G is called an n-centralizer group if #Cent(G)=n, and a primitive n-centralizer group if #Cent(G)=#Cent(G/Z(G))=n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n ≠ 2,3, there exists a finite group G with #Cent(G)=n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G)=6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) ≅ A4, the alternating group on four letters. Also, we prove that, if G/Z(G) ≅ A4, then #Cent(G)=6 or 8, and construct a group G with G/Z(G) ≅ A4 and #Cent(G)=8.

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