If G is a finite set closed under an associative operation such that ax=ay forces x=y and ua=wa forces u=w, for every a,x,y,u,w in G, prove that G is a group.
this is a good question! let then by the left cancellation property and so for some let then by associativity we
have: and thus by the left cancellation property and hence the claim is that by the right cancellation property:
thus for any there exists such that thus: now let then and so by the left cancellation property:
thus this proves that so the only thing left is to show that every element of G has an inverse. let then thus there exists such
that so every element of G has a right inverse. particularly for some but then hence and the proof is complete.