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 $gcong s_3, d_{10}$ or $s_3times s_3$. 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 collo...
