# Introduction to group theory/Problem 1 solution

Proof:

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

${\displaystyle ac=cb\implies a=b}$.

Let ${\displaystyle x,y\in G}$.

By reflexivity ${\displaystyle xyx=xyx}$.

Reassociating for clarity ${\displaystyle (xy)x=x(yx)}$.

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

${\displaystyle xy=yx}$.

Thus cross cancellation implies commutativity.

Q.E.D.