For all $ n \in \mathbb{N}-\{0,1 \}$ is there a group $G_n$ such that for all $ 1 < m \leq n$ there is $ x \in G-\{e \}$ such that $x^m = e$ ?
$e = $ neutral element in $G$.
I think its false, even for small $n$ like $n=3$ , its just a hunch.
I need some hint.
Yes, there is: just take $G_n = S_n$, and for $m \leq n$, take the cycle $(1\,2\,\ldots\,m)$.