Multiplicative group of a permutation matrix

Question

My question:

Let $M_\pi$ be the permutation matrix of an arbitrary permutation $\pi$. What is the group $\langle M_\pi \rangle$ generated by $M_\pi$ over matrix multiplication?

My effort:

I understand that it is a finite group and the number of elements in the group will be the period of $\pi$. If the period is $p$, will the multiplicative group of $M_\pi$ be the cyclic group $C_p$?

Answer 1

Consider the natural group homomorphism $\phi$ that takes a permutation to its matrix representation (to prove: this is a well-defined function and homomorphism). Then it is easy to show that $\phi$ is injective and so for $\pi \in S_n,$ we get $\langle \pi \rangle \cong \phi(\langle \pi \rangle) \cong \langle \phi(\pi)\rangle$ where $\phi(\pi)\equiv M_\pi$.