Nonempty, associative, and closed under inverses but not a group Given an example of a set $G$ and an operation $*$ on $G$ such that $*$ is not a binary operation on $G$ but associative, identity and inverses properties hold?

Basically, try to find an example to show that the closure property must be hold to be a group
How about the set $G = \{-1,0,1\}$ under usual addition?
All integers under addition. 
Let Z= all integers:
where G= (Z,+,0)
Closure property:

consider the subset {1,2} under integers, 
where 1+2=3 and 3 is an integer. Thus, proves the closure property holds for all integers under addition.