This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that:

- L
_{4}(2) and L_{3}(4) both have order 20160 - O
_{2n+1}(q) and S_{2n}(q) have the same order for q odd, n > 2

I think this means that for each integer g, there are 0, 1 or 2 simple groups of order g.

Do we need the full strength of the Classification of Finite Simple Groups to prove this, or is there a simpler way of proving it?

*(Originally asked at math.stackexchange.com).*