I was curious if anyone knew of any proofs or knew of how one might go about proving problems involving restoring permutations. An example of the type of proof I am interested in is:
Prove that any $\sigma \in S_4$ can be restored to the identity permutation, $\varepsilon$, in at most 4 moves, where a move is defined as switching two adjacent elements of $\sigma$.
I have a very basic knowledge of permutations, just what I have read from an abstract algebra book, basically consisting of cycle notation and transpositions and the like. So I know that the worst case scenario, the only case requiring 4 moves, is $\sigma = (3,4,1,2)$. I mean I know that you could do a proof by exhaustion for this example, but if you were dealing with a larger set that would not be a fun undertaking. So what I was wondering is the general method of tackling a proof like this.
I know that you could define each move as a transposition, $(1 2), (2 3), (3 4), (4 1)$, but beyond that I am just lost. Would this require some research into groups on my part?