• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.403-410
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• 2006

For a finite group G, #Cent(G) denotes the number of centralizers of its elements. A group G is called n-centralizer if #Cent(G) = n, and primitive n-centralizer if #Cent(G) = #Cent($\frac{G}{Z(G)}$) = n. The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite group G is primitive 7-centralizer if and only if $\frac{G}{Z(G)}{\simeq}D_{10}$ or R, where R is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute #Cent(G) for some finite groups, using the structure of G modulu its center.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.115-124
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• 2000

For a finite group G, #Cent(G) denotes the number of cen-tralizers of its clements. A group G is called n-centralizer if #Cent( G) = n. and primitive n-centralizer if #Cent(G) = #Cent(${\frac}{G}{Z(G)$) = n. In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with Qxactly SLX distinct centralizers. We prove that if G is a 6-centralizer group then ${\frac}{G}{Z(G)$${\cong}D_8$,$A_4$, $Z_2{\times}Z_2{\times}Z_2$ or $Z_2{\times}Z_2{\times}Z_2{\times}Z_2$.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.217-227
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• 2005

Let G be a finite group and $\#$Cent(G) denote the number of centralizers of its elements. G is called n-centralizer if $\#$Cent(G) = n, and primitive n-centralizer if $\#$Cent(G) = $\#$Cent($\frac{G}{Z(G)}$) = 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 if G is a finite group such that G/Z(G)$\simeq$$A_5$, then $\#$Cent(G) = 22 or 32. Moreover, we prove that As is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of As in terms of the number of centralizers