Scott defines group in his book "Group Theory" as follows:
Definition: A group is an ordered pair $(G,\circ)$ such that $G$ is a set, $\circ$ is an associative binary operation on $G$, and exists $e\in G$ such that
(i) if $a\in G$, then $a\circ e=a$,
(ii) if $a\in G$, then exists $a^{-1}\in G$ such that $a\circ a^{-1}=e.$
In the Exercise 1.2.16 he asks to prove that 1) if $(G,\circ)$ and $(H,\circ)$ are groups, then $G=H$. This means that the set $G$ of a group is determined by the operation $\circ$ of the group. This fact permits one to define a group as an operation $\circ$ with certain properties. Make this definition.
I am confused! I don't know what I am supposed to do here. $\circ : G\times G\rightarrow G$ so his domain is $G\times G$. So $G$ must be equal to $H$. Did I get it wrong? How to solve the last part?
Thanks for your help.