I'm confused with how a textbook presents their proof on how every permutation in a permutation group can be represented as a product of transpositions. They said the following:
"Let $\alpha \in S_n$ be a permutation and let $m$ be the number of points moved by $\alpha$. Suppose that $m = 0$. Then $\alpha = ()$. Now suppose that $m \gt 0$. Let $a$ be a single point moved by $\alpha$ and suppose that $b = a^{\alpha}$. Let $\tau = (a, b)$. Then the number of points moved by $\tau^{-1} \alpha$ is the number of points moved by $\alpha$, except for just $b$."
The only problem in this I can see is that what if $\alpha$ itself is equal to $(a,b)$? If $\alpha = (a,b)$, then $\alpha$ moves $a$ to $b$. But then $\tau^{-1} \alpha$ doesn't move $a$ at all. So what is going on here?