On finite groups with exactly seven element centralizers |
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Authors: | Ali Reza Ashrafi Bijan Taeri |
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Institution: | 1. Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran 2. Department of Mathematics, Isfahan University of Technology, Isfahan, Iran
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Abstract: | For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . 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 groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR 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 ofG modulu its center. |
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