Quote by Xil3
First when trying to show isomorphism between groups is it enough to show that the order of each element within the group is the same in the other group?

No.
And does the binary operator have to be the same in both groups when doing group isomorphism?

What do you mean by 'same'? If f is an isomorphism then f(xy)=f(x)f(y), if that's what you mean.
Is there a missing 1 there?
I know that this ring has 4 elements. Is it correct that the elements are the following
1
x + 1
x^2 + 1
x^2 + x + 1

Where is 0? Why haven't you written down x or x^x+x? (You have, by the way, but you have made a strange choice of representatives of the elements, which is why I ask, since it implies you've not really understood what you're doing.