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One way to prove the Parity Theorem is first to prove the

** Even Identity Lemma.** If e is the identity permutation in

Before considering a proof of this lemma, let us show how it leads very directly to a proof of the Parity Theorem. Suppose that we have two expressions, and , for a permutation a in terms of transpositions. Then, since the inverse of a composition of a sequence of permutations is the composition of their inverses in the reverse order, and since every transposition is its own inverse, it follows that

This shows that e can be expressed using *k + m* transpositions. Once the lemma is established, we will know that *k + m* is an even number. This assures that *k* and *m* are either both even numbers or they are both odd numbers.

John O. Kiltinen, " Parity Theorem for Permutations - An Enabling Lemma," *Loci* (December 2004)

Journal of Online Mathematics and its Applications