Prove that if G is a finite group and whose index is the least prime dividing the order of G, then .
Proof:
This is all I could come up with-
, , and let .
Now, and .
Now what do I do?
Here is the classic way to do this problem. Let act on (the coset space, not the quotient group) in the usual way. This gives a homomorphism . It's trivial that . That said, note that since divides it cannot be divisible by any smaller prime than yet since divides (by FIT) it clearly then follows that and so . Since this implies that and so .
A more interesting proof, if you know double cosets, can be found here on my blog.