# Group has Latin Square Property/Additive Notation

## Theorem

Let $\struct {G, +}$ be a group.

Then $G$ satisfies the Latin square property.

That is, for all $a, b \in G$, there exists a unique $g \in G$ such that $a + g = b$.

Similarly, there exists a unique $h \in G$ such that $h + a = b$.

## Proof

From Group has Latin Square Property, we have that:

 $\ds a + g$ $=$ $\ds b$ $\ds \leadsto \ \$ $\ds g$ $=$ $\ds \paren {-a} + b$

and:

 $\ds h + a$ $=$ $\ds b$ $\ds \leadsto \ \$ $\ds h$ $=$ $\ds b + \paren {-a}$

$\blacksquare$