I'm writing in reference to the answer to the question: If $\epsilon = \beta_1 \beta_2 ...... \beta_r$ where the $\beta's$ are 2- cycles, then r is even.
To be honest, even after looking after the answer a few things are unclear to me.
I don't understand the use of expressing $\epsilon$ in the forms:
$\epsilon = (ab)(ba)$,
$(ab)(bc)=(ac)(bc)$,
$(ac)(cb)=(bc)(ab)$,
$(ab)(cd)=(cd)(ab)$
Could someone tell me how he came up with these random forms of permutations (for the last two positions)? Is there any logic involved which I'm missing?
If $\beta_{r-1}\beta_r$ is in the first form I understand that that would be equal to the identity permutation and in that case we can just omit $\beta_{r-1}\beta_r$. BUT, how can we conclude from that $r-2$ is even, from there? Gallian says that it is by "Second Principle of Mathematical Induction". No idea what that means.
Gallian says that in the other three cases we need to replace the form of $\beta_{r-1}\beta_r$ on the right by "its counterpart on the left"? What does "counterpart" mean in this context? (I'm sorry if that's a silly question, but English isn't my first language, so I might be having trouble interpreting)
Moreover, what does he mean by "to obtain a new product of $r$ 2-cycles that is still the identity, but where the rightmost occurrence of the integer $a$ is in the second-from-the-rightmost 2-cycle". We now repeat the procedure just described with $\beta_{r-2}\beta_{r-1}$ and as before we obtain a product of $(r-2)$ 2-cycles equal to the identity or a new product of $r$ $2$-cycles where the rightmost occurrence of $a$ is in the third $2$-cycle from the right"?