The first homology group coincides with abelianization, and (since A_n is perfect, except for the cited exceptions) is thus:
:H₁(A_n, Z) = Z₁ for n = 0, 1, 2;
:H₁(A₃, Z) = A = A₃ = Z₃;
:H₁(A₄, Z) = A = Z₃;
:H₁(A_n, Z) = Z₁ for n >= 5.

This is easily