Proving a group G is isomorphic to D_10

[alex07966]

Question: Let G be a non-abelian group such that the size of G is 10. Prove G is isomorphic to D_10.

I have started by saying the elements of G have order 1,2,5 or 10. And then showing how there are no elements of order 10 as that would make it abelian. I then show there is an element x of order 5 and an element y of order 2.
I then show G = {e, x, x^2, x^3, x^4, y, xy, (x^2)y, (x^3)y, (x^4)y} using right cosets.
NOW... i need to find what yx is equal to:
I then show that yx must be equal to either (x^2)y, (x^3)y or (x^4)y.
I think i then need to show BY CONTRADICTION that yx is not equal to the first two elements and therefore must be equal to (x^4)y. I can't reach a contradiction though! please help.

I can then say this group G has the same equations for working out the multiplication table of D_10 and is therefore isomorphic to D_10.

 

[Billy Bob]

Here's one way. Write $$yxy^{-1}=x^m$$.

Try to obtain $$x=y^2 xy^{-2}=(x^m)^m$$ and thus $$m^2$$ is congruent to 1 modulo 5.