Show that $\operatorname{sgn}(\sigma^n)=(-1)^n$ for specific permutation in $S_7$.

Question

Let $\sigma = \begin{pmatrix} 1 &2&3&4&5&6&7\\ 4&5&7&2&1&3&6 \end{pmatrix}\in S_7$.

Show that $\operatorname{sgn}(\sigma^n)=(-1)^n$.

So I know that $\operatorname{sgn}(\sigma)=\operatorname{sgn}((4,2,5,1))\operatorname{sgn}((7,6,3))=(-1)^3(-1)^2=-1$, but I'm not sure how to continue. Any help is appreciated

Answer 1

Now you use the fact that the sign function is a group homomorphism. So, $\operatorname{sgn}(\sigma^n)=\operatorname{sgn}(\sigma)^n$.