Let $F$ be a finite field, and let $P, D\in GL(n,F)$, where $P$ is a permutation matrix and $D$ is a diagonal matrix.
Let $M=PD$. Let $o(M)$ denote the multiplicative order of $M$; $o(M)$ is the smallest element of the set $\{n\in\mathbb{Z}^+ \vert M^n=I\}$. My question is: what are some useful formulae for $o(M)$?
Thoughts:
Let $\mathbf{v}_1\dots,\mathbf{v}_n$ be the standard basis for an $n$-dimensional vector space $V$ over $F$. That is to say, $\mathbf{v}_i=\begin{pmatrix}{0,\dots,0,1,0, \dots, 0}\end{pmatrix}$, you know how it goes. Then $P$ permutes the one-dimensional subspaces $\{\langle v_i\rangle\}_{1\leq i\leq n}$, while $D$ fixes these.
If we consider the action by $M$ on these one-dimensional subspaces, then we can see that $M^{o(P)}$ is a diagonal matrix.
Since the multiplicative order of a diagonal matrix must divide $|F^\times|$, this means that $o(M)$ divides $|F^\times|o(P)$.
More thoughts:
Let $\Omega=\{1,\dots, n\}$, and let $\sigma\in\text{Sym}(\Omega)$ be the permutation of $\Omega$ that corresponds to $P$.
Let $a_i$ be the $i$-th diagonal element of $D$. Let $m=o(P)=o(\sigma)$.
Then $(M^{o(P)})_{ii}=\prod_{1\leq j < m}a_{\sigma^j(i)}$