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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.

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What is $s\star s^{n_s-1}$? –  Daniel Rust Mar 31 at 1:16
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I changed the title to something useful. Regards –  rschwieb Mar 31 at 2:20

1 Answer 1

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 $s⋆(s^{N-1}) = e$. Given this, what is the inverse of our element $s$?

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