# 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

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Example $84$