Finite groups have even more centralizers

Document Type : Research Paper


Department of Mathematics‎, ‎Faculty of Sciences‎, ‎University of Zanjan‎, ‎P.O‎. ‎Box 45371-38791‎, ‎Zanjan‎, ‎Iran


For a finite group $G$‎, ‎let $Cent(G)$ denote the set of centralizers of single elements of $G$‎. ‎In this note we prove that if $|G|\leq \frac{3}{2}|Cent(G)|$ and $G$ is 2-nilpotent‎, ‎then $G\cong S_3‎, ‎D_{10}$ or $S_3\times S_3$‎. ‎This result gives a partial and positive answer to a conjecture raised by A‎. ‎R‎. ‎Ashrafi [On finite groups with a given number of centralizers‎, ‎Algebra‎ ‎Colloq. 7 (2000)‎, ‎no‎. ‎2‎, ‎139--146]‎.


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