Journal of Applied Mathematics and Computing

, Volume 22, Issue 1–2, pp 403–410

# On finite groups with exactly seven element centralizers

Article

## 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.

20D99 20E07

### Key words and phrases

Finite group n-centralizer group primitiven-centralizer group

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