Permutation as a product of transposition

Question

I'm trying to figure out how the proof of the following theorem works: THEOREM: Every permutation is a product of transpositions. The proof is based on noetherian induction. I don't understand how it goes: proof for the number $m$ of the numbers $i$ from {$1,2,...,n$} for which $\sigma(i) \neq i$. How does this suffice? Why exactly this connection with fixed points? Thank you very much!

Answer 1

For a permutation $j_1 j_2 \cdots, j_n$

Imagine people from 1 to $n$ sitting on seats with the same numbers, when one person moves, he changes his seat with one person beside him.

Then person $j_n$ moves to seat $n$ by crossing all the persons in front of him, then person $j_{n-1}$ moves to seat $n-1$ in the same way, etc.

A move is a transposition, and at the end we have the permutation $j_1 j_2 \cdots, j_n$