The first homology group is the abelianization, and corresponds to the sign map S n → S 2 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 S n → C p are to S 2 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 S n → S 2 ≅ {±1} 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.