91Ó°ÊÓ

Show that a power of a cycle need not be a cycle.

(i) Let \(\alpha=\beta \gamma\) in \(S_{n}\), where \(\beta\) and \(\gamma\) are disjoint. If \(\beta\) moves \(i\), then \(\alpha^{k}(i)=\beta^{k}(i)\) for all \(k \geq 0\). (ii) Let \(\alpha\) and \(\beta\) be cycles in \(S_{n}\) (we do not assume that they have the same length). If there is \(i_{1}\) moved by both \(\alpha\) and \(\beta\) and if \(\alpha^{k}\left(i_{1}\right)=\beta^{k}\left(i_{1}\right)\) for all positive integers \(k\), then \(\alpha=\beta\).

Let \(p\) be a prime and let \(\alpha \in S_{n}\). If \(\alpha^{p}=1\), then either \(\alpha=1, \alpha\) is a \(p\) -cycle, or \(\alpha\) is a product of disjoint \(p\) -cycles. In particular, if \(\alpha^{2}=1\), then either \(\alpha=1, \alpha\) is a transposition, or \(\alpha\) is a product of disjoint transpositions.

Show that \(S_{n}\) has the same number of even permutations as of odd permutations. (Hint. If \(\tau=(12)\), consider the function \(f: S_{n} \rightarrow S_{n}\) defined by \(f(\alpha)=\tau \alpha\).)