# Group has Latin Square Property/Corollary

## 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 $a$.

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

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.

$\blacksquare$

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 5$: The Multiplication Table - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.4$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(v)}$