I'm stuck on this problem. I really have no clue. Would someone give me a hand?

1)Show $\displaystyle \forall i$ $\displaystyle \alpha \in S_n$ moves $\displaystyle i$ iff $\displaystyle {\alpha}^{-1}$ moves $\displaystyle i$

2) Show that if $\displaystyle \alpha \beta=(1)$ and $\displaystyle \alpha , \beta$ are disjoint, then $\displaystyle \alpha=(1)$ and $\displaystyle \beta=(1)$