I am studying group representations and to prove that characters from symmetric groups $\chi(g) \in \mathbb{Z}, \forall g \in S_{n}$ I need prove that:
Consider that $\sigma \in S_{n}$, and $\gcd(m, o(\sigma)) = 1$. Then $\sigma$ and $\sigma^{m}$ has the same cycle structure.
Someone can help me with this question. Thank you in advance.
$\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$If $\tau$ is a cycle of length $k$ in the disjoint cycle decomposition of $\sigma$, then $k \mid o(\sigma)$.
You have thus to prove that if $\gcd(m, k) = 1$, then $\tau^{m}$ is also a cycle of length $k$.
Consider the cyclic subgroup $\Span{\tau}$ of $S_{n}$. Then a standard result shows that $o(\tau^{m}) = o(\tau)$, so that $\Span{\tau} = \Span{\tau^{m}}$. In particular, $\tau$ is a power of $\tau^{m}$. This implies that $\tau^{m}$ is a cycle of length $k$.