If I've got a group $G$. Is it possible for $G$ having only one element from order $5$?
I know that for example if we take $G$ as a cyclic group of order $n$, $G$ must have only one subgroup for each divisor of $n$. But what happens with the elements? Is it possible?
Best Answer
If $g\in G$ has order $5$, then the subgroup $H=\langle g\rangle$ generated by $g$ is cyclic of order $5$. All non-trivial elements of $H$ (hence at least four elements of $G$) have order $5$.