If $(\mathbb Z,*)$ is invertible and has cancellation property, does it imply that is associative?

Question

If $(\mathbb Z,*)$ is invertible and has cancellation property, does it imply that is associative?

I know that if is invertible and has associative, then is a group and has cancellation property. But I'm confused, Can I say it is associative, only known that is invertible and has cancellation?

Answer 1

Any kind of non-associative loop where each element has an inverse will do as a counter-example. This includes all non-associative Moufang loops, such as the non-zero octonions.