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# Cancellation and Neutral Elements in Groups: Solving Equations

ABSALG-SYTBG9

Let $G$ be a group with composition law $\circ$.

The property $a\circ x=x\circ a=x$ for all $a\in G$

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For $a\in G$, the equation $a\circ x=a\circ y$

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For all $a\in G$, the equation $x\circ a=y\circ a$

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For all $a\in G$, the equation $x\circ a\circ u=y\circ a\circ v$