Introduction to group theory/Problem 1 solution

Proof:

Let $G$ be a group such that for all $a,b,c\in G$

$ac=cb⟹a=b$.

Let $x,y\in G$.

By reflexivity $xyx=xyx$.

Reassociating for clarity $(xy)x=x(yx)$.

By the assumed cross cancellation we may cancel on each side to obtain

$xy=yx$.

Thus cross cancellation implies commutativity.

Q.E.D.