I'm trying to prove that is not isomorphic to (the notation means ) but I'm having difficulties when I get to the final conclusion.

This is my work so far (from now on, I'll drop the and notations for conveniency):

Suppose that is an isomorphism (I'll try to derive a contradiction).

Then it must hold that .

Let , then it has its inverse .

So:

Finally:

(properties of multiplication in )

Now, since we know that can be either or ... what does that tell me? that is not well defined (and therefor cannot be isomorphism)? ( is sent to two different ...)