Izaak Meckler
Is there a simple proof that every faithful rep of $$S_n$$ has dimension at least $$n - 1$$?

The only proof I know that every faithful rep of $$S_n$$ has dimension at least $$n - 1$$ goes like this:

1. Write down the irreducible representations of $$S_n$$.
2. Observe that you wrote a number of reps equal to the number of conjugacy classes and so conclude that you found them all.
3. Observe that the smallest dimensional faithful irrep has dimension $$n - 1$$.
4. The non-faithful representations can only see the sign, and so don't help in building a faithful rep, thus every faithful rep contains a faithful irrep and so has dimension at least $$n - 1$$.

Under some reasonable(?) complexity-theoretic assumption, we can give a really simple proof that any faithful rep has dimension $$\Omega(n^c)$$ for some constant $$c > 0$$. Is there a simple unconditional proof?