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Let $G$ be a finite, nonempty set with an operation $*$ such that
- $G$ is closed under $*$ and $*$ is associative
- Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$.
- Given $a,b,c \in G$ with $b*a=c*a$, then $b=c$.
I want to prove that $G$ is a group, but I don't know how to show that there exists an identity $e\in G$ such that $e*x=x$ and $x*e=x$ $\forall x \in G$. I also don't know how to show that $\forall$ x $\in G$ there exists a $y \in G$ such that $y*x=e$ and $x*y=e$. How do I do this?