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:
\(\ds a + g\) | \(=\) | \(\ds b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds g\) | \(=\) | \(\ds \paren {-a} + b\) |
and:
\(\ds h + a\) | \(=\) | \(\ds b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds h\) | \(=\) | \(\ds b + \paren {-a}\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Example $84$