My problem is as follows: Let H = { e, x, y, z) be a group under the binary operation ∆. e is the identity element and all elements are different from each other. I have to prove that x∆y ≠ z∆x my idea was to assume wrongly that x∆y = z∆x and then to reach some sort of a contradiction such as a = b or c = e or something contradicting and by that completing the prof. I believe that by applying some "cosmetic" changes such as adding * e to one side or another I could reach that, but so far I have been unlucky. If anybody could show me the way I would greatly appreciate it.
kind regards
Suppose that $x\Delta y =z\Delta x$. We can't have these expressions equal to $e$, for then both $y$ and $z$ would be inverses of $x$, so they would be equal. And they can't be equal to $y$ or $z$, since, for instance, $z\Delta x = z \implies x=e$. Finally, if these expressions equal $x$, then $x\Delta y = x \implies y=e$. Therefore, we reach a contradiction in all possible cases.