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Paper   IPM / M / 8011
School of Mathematics
  Title:   Groups with specific number of centralizers
  Author(s):  A. Abdollahi (Joint with S. M. Jafarian Amiri and A. Mohammadi Hassanabadi)
  Status:   Published
  Journal: Houston J. Math.
  Vol.:  33
  Year:  2007
  Pages:   43-57
  Supported by:  IPM
Let G be a group and let cent (G) denote the set of centralizers of single elements of G. A group G is called ncentralizer if |cent (G)|=n. In this paper, for a finite group G, we give some interesting relations between |cent (G)| and the maximum number of the pairwise non-commuting elements in G. Also we characterize all ncentralizer finite groups for n=7 and 8. Using these results we prove that there is no finite group G with the property that |cent (G)|=|cent([(G)/(Z(G))])|=8, where Z(G) denotes the centre of G. This latter result answers positively a conjecture posed by A. R. Ashrafi.

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