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- Apr 14, 2013

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I am looking at the following exercise:

Make a sketch of a regular tetrahedron and label the corners with the numbers $1, 2, 3, 4$. For $1\leq i\leq 5$ the permutations $\pi_i \in \text{Sym} (4)$ are defined as follows:

\begin{align*}&\pi_1:=\text{id} \\ &\pi_2:=(1 \ \ 2) \\ &\pi_3:=(1 \ \ 2 \ \ 3) \\ &\pi_4:=(1 \ \ 2)\ (3 \ \ 4) \\ &\pi_5:=(1 \ \ 2 \ \ 3 \ \ 4)\end{align*}

- For $1 \leq i \leq 5$, check whether there is a symmetry of the tetrahedron that the permutation $\pi_i$ induces on the set of vertices $\{1, 2, 3, 4\}$ and, if so, give a geometric description of this symmetry.
- Is there for every $\pi \in \text{Sym} (4)$ a symmetry of the tetrahedron, that induces the permutation $\pi$ on the set of vertices $\{1, 2, 3, 4\}$ ?

Could you explain to me the exercise and what I am supposed to do?

I got stuck what it means "induce on a set".