The forward implication is trivial.

Suppose then that for any .

Need to show closure under the operation, associativity, that identities exist, and that an identity element exists.

Well since (it's nonempty remember), the product by the given conditions. Do you see how this impliesGcontains the identity element (call ite)?

Since and are inG, then the product . Do you see how this impliesGcontains inverses for each of its elements?

All you have to do now is acknowledge the inheritance of the associativity property and show that the operation is closed inG.

If you pick what gentle subtlety do you need, using what you know, so you can show ?