Group has Latin Square Property/Additive Notation

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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:

\(\displaystyle a + g\) \(=\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle g\) \(=\) \(\displaystyle \paren {-a} + b\)

and:

\(\displaystyle h + a\) \(=\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle h\) \(=\) \(\displaystyle b + \paren {-a}\)

$\blacksquare$


Sources