The first homology group is the abelianization, and corresponds to the sign map S"n" → S2 which is the abelianization for "n" ≥ 2; for "n" < 2 the symmetric group is trivial. This homology is easily computed as follows: S"n" is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps are to S2 and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S"n".