In this book I am reading, a group is defined as:
A group is an ordered pair $(G,X)$ where $G$ is a set and $X$ is a binary operation on $G$ satisfying the following axioms:
i) $X$ is associative.
ii) $G$ contains an identity.
iii) Each element of $G$ has an inverse.
How does this definition imply that $G$ is closed under this operation?