Futurama Theorem

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

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:


 * $(1): \quad y x = z$, where $z$ contains no transpositions from $H$
 * $(2): \quad y_a \ne x_b$ for all $a, b$.

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.