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?