In theory, I could go back and rewrite these notes, introducing the reader first to lists, then to permutations, then to S2, to S3, to the subgroups of S4 that correspond to the cyclic group of order four and the Klein 4-group, and so forth, not refer to these other groups, nor to the dihedral group, nor to any other finite group that we have studied.
But it is more natural to think in terms other than permutations (geometry for Dn is helpful), and it can be tedious to work only with alterations. While Cayley’s Theorem has its uses, it does not suggest that we should always consider groups of permutations in place of the more natural representations in scientific notation.
With that in mind, consider two different representations of α by transpositions. If the first representation has an even number of transpositions, say α = τ1 ···τ2a, then we can determine gα by applying each τ one after the other. We see that swp α = swap τ1 +···+swp τ2a, an even sum of odd numbers, which is even. Hence gα = g . If the second representation had an odd number of transpositions, then swp α would be an odd sum of odd numbers, and gα = −g.
However, we pointed out in the previous paragraph that the value of gα depends on the permutation α, not on its representation by transpositions. Since g is non-zero, it is impossible that gα = g and gα = −g. It follows that both representations must have the same number of transpositions. The statement of the theorem follows: if we can write α as a product of an even (resp. odd) number of transpositions, then it cannot be written as the product of an odd (resp. even) number of transposition.