Introduction to group theory/Problem 1 solution


 * Proof:

Let $$G$$ be a group such that for all $$a,b,c \in G$$
 * $$ac=cb \implies a=b$$.

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.