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.