Let be a field. Prove that as groups, and are not isomorphic.
First suppose is finite. Then and for some natural number . In particular, no bijection exists, so that these groups cannot be isomorphic.
Suppose now that is infinite and that is a field isomorphism. Note that .
Suppose . Now for some , and . But also . So . Thus , and we have . But then , and is not injective, a contradiction.
Suppose instead that . Now for some , so that . Since is injective, . Thus , and we have , a contradiction.
Thus and are not isomorphic.