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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.
Algebra Colloquium – Springer Journals
Published: Jan 1, 2000
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