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.

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