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- Apr 14, 2013

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1. Let $1\leq n\in \mathbb{N}$ and $\pi\in \text{Sym}(n)$. For $1\leq k\in \mathbb{N}$ we define $\pi^{-k}:=\left (\pi^n\right )^{-1}$.

Show for all $k,\ell\in \mathbb{Z}$ the equation $\pi^k\circ \pi^{\ell}=\pi^{k+\ell}$.

2. Let $1\leq n\in \mathbb{N}$. Show that $\pi^{n!}=\text{id}$ for all $\pi\in \text{Sym}(n)$.

Do we show both statements using induction?