The proof on page 103 in Gallian’s 9th edition all makes sense to me except how he concludes that the identity permutation must be equal to $r-2$ $2$-cycles assuming the other three cases. His entire proof is mentioned in this question. Specifically my confusion begins where it says “Continuing…”
I’m also not sure why it’s not just mentioned that the identity could be written in $r-2k$ $2$-cycles where $k$ is how many times the $2$-cycles repeated and cancelled out while filtering the permutation. In other words, why did the filtering process end up canceling exactly two $2$-cycles?
To be clear, I understand why the identity can’t be a product of $r$ $2$-cycles and how it leads to a contradiction. My problem is with how Gallian makes it seem like the identity permutation has to be a product of $r-2$ $2$-cycles.