A note on cut groups of odd order

Abstract

A group $G$ is said to be cut if, for every $g\in G$, each generator of $<g>$ is conjugated to either $g$ or $g^{-1}$. It is conjectured that a Sylow 3-subgroup $P$ of a cut group $G$ is cut. We prove that this is true if $|G|$ is odd.