Permuted indices of a monomial

Question

Consider the monomial

\begin{align} M_{\gamma^{-1}(1),\sigma^{-1}(1)} M_{\gamma^{-1}(2),\sigma^{-1}(2)} \dots M_{\gamma^{-1}(n),\sigma^{-1}(n)}, \end{align}

where each $M_{i,j}$ an entry of a real matrix and $\gamma,\sigma \in S_n$. I want to write the monomial in the form

\begin{align} M_{\pi(1),1}M_{\pi(2),2} \dots M_{\pi(n),n} \end{align}

where $\pi$ is some permutation written in terms of $\gamma$ and $\sigma$. I've looked at the small cases and have found that $\pi=\gamma^{-1}\sigma$. I want to be sure that this holds in general.

Answer 1

....figured it out. There exists an index $i$ such that $\sigma^{-1}(i)=1$ and hence $\sigma(1)=i$. We get

\begin{align} M_{\gamma^{-1}(i),\sigma^{-1}(i)} = M_{\gamma^{-1} \sigma(1),1}. \end{align}

Hence $\pi = \gamma^{-1}\sigma$.