Futurama Theorem

Theorem
Let $A_{n-2}\subset A_n \ $ be a subgroup of $A_n \ $, the alternating group on n letters.

For any element $x\in A_{n-2} \ $, let $x=x_1 x_2 \dots x_k \ $, where $x_i \in H \ $ is a transposition.

Then there exists $y \ $ which can be represented as a series of transpositions $y_1 y_2 \dots y_j \in A_n \ $ such that:


 * $yx = z \ $, where $z \ $ contains no transpositions from $H \ $,
 * and $y_a \neq x_b \ $ for any $a, b \ $.

Proof
Let $w = (n[n-1]) \ $, that is, the transposition of the $n^{th} \ $ and $(n-1)^{th} \ $ letters that we consider $A_n \ $ acting on.

Then the permutation $x^{-1}w \ $ is the $y \ $ of the theorem.