State/Cancellation in groups

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Proposition: (Cancellation in groups) Suppose that $G$ is a group, and $g,x,y \in G$. Then, if $gx = gy$, then $x = y$. Similarly, if $xg = yg$, then $x = y$.

Logical Connections

This statement logically relies on the following definitions and statements: Def/Group

The following statements and definitions rely on the material of this page: Def/Group homomorphism, and State/Lagranges theorem

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State/Cancellation in groups

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	Suppose that $(G, \cdot)$ is a group, and $g,x,y \in G$. Suppose that $gx = gy$.
	Since $G$ is a group, $g$ has an inverse, $g^{-1}$, satisfying $g^{-1} g = 1$. Therefore, since $gx = gy$, we find that $g^{-1} (g x) = g^{-1} (g y)$. By the associative property, we find that $(g^{-1} g) x = (g^{-1} g) y$. Hence $1 \cdot x = \cdot y$.
	Therefore $x = y$.