Group has Latin Square Property/Proof 3
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Theorem
Let $\struct {G, \circ}$ 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 $a \circ g = b$.
Similarly, there exists a unique $h \in G$ such that $h \circ a = b$.
Proof
Suppose that $\exists x, y \in G: a \circ x = b = a \circ y$.
\(\ds a \circ x\) | \(=\) | \(\ds a \circ y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^{-1} \circ \paren {a \circ x}\) | \(=\) | \(\ds a^{-1} \circ \paren {a \circ y}\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a^{-1} \circ a} \circ x\) | \(=\) | \(\ds \paren {a^{-1} \circ a} \circ y\) | Group Axiom $\text G 1$: Associativity | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds e \circ x\) | \(=\) | \(\ds e \circ y\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds y\) | Group Axiom $\text G 2$: Existence of Identity Element |
So such an element, if it exists, is unique.
Now it is demonstrated that $g = a^{-1} b$ satisfies the requirement for $a \circ g = b$
Since $a \in G$, it follows by group axiom $G3$: existence of inverses that $a^{-1} \in G$.
\(\ds a\) | \(\in\) | \(\ds G\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^{-1}\) | \(\in\) | \(\ds G\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^{-1} \circ b\) | \(\in\) | \(\ds G\) | Group Axiom $\text G 0$: Closure |
Then:
\(\ds a \circ g\) | \(=\) | \(\ds a \circ \paren {a^{-1} \circ b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ a^{-1} } \circ b\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds e \circ b\) | Group Axiom $\text G 3$: Existence of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds b\) | Group Axiom $\text G 2$: Existence of Identity Element |
Thus, such a $g$ exists.
The properties of $h$ are proved similarly.
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: $(1.9)$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Proposition $3.3$