Show that $\alpha^m = \varepsilon$ using $\alpha^\ell (a_i) = a_{(i+\ell) \bmod{m}}$ where $\alpha = (a_0 a_1 \dots a_{m-1}) \in S_n$ a permutation group.
I've been working on this problem but can't seem to get anywhere. Any direction is always appreciated
Since the group of permutations of lenght n is finite there exists n and p for which $a^n = a^p $ with n > p. By multiplying with $a^{-p}$ we have $a^{n-p} = e$.