This question pertains to the proof mentioned in this question. What is the inductive hypothesis here? As I can decipher it, it seems like:
If $i$ is the number of terms in 2-cycle identity permutations and $i<r$ then $i$ is even.
Can someone verify this? Or if wrong mention the correct hypothesis.
This is an example of what is sometimes called strong induction. The hypothesis of strong induction is that the theorem is true for all values less than $r$. One place you see the strong induction hypothesis used is where it says "By induction, $r-2$ is even; hence, $r$ must be even." Your statement does, indeed, appear to be identical to the strong induction hypothesis.