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 implies G contains the identity element (call it e)?
Since and are in G, then the product . Do you see how this implies G contains 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 in G.
If you pick what gentle subtlety do you need, using what you know, so you can show ?