# 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

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How about the set $G = \{-1,0,1\}$ under usual addition?