We show how the fusion rules for an affine Kac-Moody Lie algebra g of type
A_{n-1}, n = 2 or 3, for all positive integral level k, can be obtained from
elementary group theory. The orbits of the kth symmetric group, S_k, acting on
k-tuples of integers modulo n, Z_n^k, are in one-to-one correspondence with a
basis of the level k fusion algebra for g. If [a],[b],[c] are any three orbits,
then S_k acts on T([a],[b],[c]) = {(x,y,z)\in [a]x[b]x[c] such that x+y+z=0},
which decomposes into a finite number, M([a],[b],[c]), of orbits under that
action. Let N = N([a],[b],[c]) denote the fusion coefficient associated with
that triple of elements of the fusion algebra. For n = 2 we prove that
M([a],[b],[c]) = N, and for n = 3 we prove that M([a],[b],[c]) = N(N+1)/2. This
extends previous work on the fusion rules of the Virasoro minimal models
[Akman, Feingold, Weiner, Minimal model fusion rules from 2-groups, Letters in
Math. Phys. 40 (1997), 159-169].