> However, you can't infer S(2) is true from assuming S(1) is true in the same way, ie. a group of 2, could be represented as two groups of S(1) and S(1).

Yup, I think this nails it. The more I think about this problem the more it seems meaningless to me. Adding the whole induction stuff just obfuscates this core problem.

It seems to me that if you can prove S(2) you've immediately proven S(n). You don't need the subsequent induction.