Let $S,\star$ be an associative binary structure with identity $e$. Assume that for every $s\in S$ there is an integer $n_s>0$ such that $s^{n_s}=e$. Show that $S,\star$ is a group.
I am not sure what to do for this problem. Any feedback would be appreciated.
Most properties of groups are presupposed in the problem statement. The important thing we need to establish is the existence of inverses.
We know that, for every $s$ in the structure, there exists an $N$ with the property that $s^N = e$. So, with a little rearranging, we have $sā(s^{N-1}) = e$. Given this, what is the inverse of our element $s$?
