Group has Latin Square Property/Corollary

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Corollary to Group has Latin Square Property

The Cayley table for any finite group is a Latin square.

Proof 1

Follows directly from the definition of both a Cayley table and a Latin square.

Proof 2

Let $G$ be a finite group whose order is $n$.

Let $\tuple {x_1, x_2, \ldots, x_n}$ be the elements of the underlying set of $G$ in the order they appear in the headings of the Cayley table of $G$.

Consider the row of the Cayley table headed with $a$.

The elements of that row are:

$\tuple {a x_1, a x_2, \ldots, a x_n}$

that is:

$\tuple {\map {\lambda_a} {x_1}, \map {\lambda_a} {x_2}, \ldots, \map {\lambda_a} {x_n} }$

where $\lambda_a$ denotes the left regular representation of $\struct {S, \circ}$ with respect to $a$.

From Regular Representations in Group are Permutations, it follows that each of $\map {\lambda_a} {x_1}, \map {\lambda_a} {x_2}, \ldots, \map {\lambda_a} {x_n}$ appears in that row exactly once.

A similar argument based on the right regular representation proves the result for the columns.