I'm trying to solve this question: Show $\pi \in S_n$ is an automorphism over $G = (V, E)$ with $V = \{1,...,n\}$ if there is an edge-set $E'$ such that: $$E = \bigcup_{i=0}^{\binom{n}{2}} \{ \pi^i(u) \pi^i(v)\ | \ uv \in E' \}$$ But I'm having a bit of trouble understanding how I would go about doing it!
Thanks for any help.
Just some partial thoughts.
We have usually three steps when proving that something is an Automorphism.
1. It is endomorphism So you ask: if $uv$ edge, is $\pi(u)\pi(v)$ edge? In decomposition of $E$, situation $i=1$ gives you this condition for edges in $E'$. Can you then extend it for the whole $E$?
2. It is surjective Decomposition gives you that any edge can be written as $\pi^i(u)\pi^i(v)$ for some $i$ and $uv\in E'$. From this follows that each edge has a pre-image and moreover we know that this pre-image is of form $\pi^{i-1}(u)\pi^{i-1}(v)$
3. It is injective Once you have proven this, your proof is complete. (Sorry for not completing the proof)