Graph automorphism group conjugacy classes

Question

Let $G$ be the automorphism group on a graph $\Gamma(V,E)$. Why is it that group elements belonging to the same conjugacy class have the same cycle type? For example, if $a,b\in G$ are two automorphisms on $\Gamma$, such that $g^{-1}ag=b$ for some $g\in G$, then $a,b$ have the same cycle type?

Answer 1

This is because $$\sigma (a_1\ \ldots \ a_n)(b_1\ \ldots \ b_k)\sigma^{-1}=\sigma (a_1\ \ldots \ a_n)\color{red}{\sigma^{-1}\sigma}(b_1\ \ldots \ b_k)\sigma^{-1}$$ and $$\sigma (a_1\ \ldots \ a_n)\sigma^{-1}=(\sigma(a_1)\ \ldots \ \sigma(a_n))$$