Let be an abelian group (written multiplicatively), . Denote by the product of all elements of . Consider the subgroups and . Prove:
(b) If is a subgr of the multiplicative group of a field and , then .
Up to here, the statement of the problem. And it is in the statement, point (b), that I find a counter-example: For the statement seems to assert n is an even number. However, let be finite, and take to be the whole multiplicative group of . Then even and is odd. Yet, , and therefore belong to G, as the author of the problem requires (that is, and not equal . So, ).
Am I right? Is the statement wrong? Any hint will be greatly appreciated. Good bye.