Okay, one last question this semester--pretty stressed out, and can't really figure it out.
We need to show that if $G$ is a doubly transitive subgroup of $S_n$ (that is, for $(x,y)$ distinct, $(u,v)$ distinct, there is some $g$ such that $g(x) = u$ and $g(y) = v$) and if $G$ contains a 3-cycle, then $G = A_n$ or $G = S_n$.
Really I'm just looking for hints in the right direction. Right now I have the intuition that I can pick any two elements and map them anywhere by the doubly-transitive property. And I think we can use that to produce all the 3-cycles, given that the group contains a 3-cycle. And that means we have at least $A_n$. This seems to be a bit of an ad-hoc way of doing it.