ON FINITE GROUPS WITH SPECIFIC NUMBER OF CENTRALIZERS

Abstract

For a group G, | Cent(G) | denotes the number of distinct centralizers of its elements. A group G is called n-centralizer if | Cent(G) |= n, and primitive n-centralizer if | Cent(G) |=| Cent(GZ(G)) |= n. In this paper, among other things, we investigate the structure of finite groups of odd order with | Cent(G) |= 9 and prove that if |G| is odd, then | Cent(G) |= 9 if and only if GZ(G)∼= C7 o C3 or C7 × C7.