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On Finite Groups with a Given Number of Centralizers

On Finite Groups with a Given Number of Centralizers 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) ≅ A 4, the alternating group on four letters. Also, we prove that, if G/Z(G) ≅ A 4, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) ≅ A 4 and #Cent(G) = 8. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Colloquium Springer Journals

On Finite Groups with a Given Number of Centralizers

Algebra Colloquium , Volume 7 (2) – Jan 1, 2000

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2000 by Springer-Verlag Hong Kong
Subject
Mathematics; Algebra; Algebraic Geometry
ISSN
1005-3867
eISSN
0219-1733
DOI
10.1007/s10011-000-0139-5
Publisher site
See Article on Publisher Site

Abstract

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) ≅ A 4, the alternating group on four letters. Also, we prove that, if G/Z(G) ≅ A 4, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) ≅ A 4 and #Cent(G) = 8.

Journal

Algebra ColloquiumSpringer Journals

Published: Jan 1, 2000

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