Hi:

Let

be an abelian group (written multiplicatively),

. Denote by

the product of all elements of

. Consider the subgroups

and

. Prove:

(a)

.

(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.