An associative Latin square is a group. A very short proof

TL;DR

This paper proves that any associative Latin square corresponds to a group, establishing a fundamental link between combinatorial designs and algebraic structures.

Contribution

It provides a concise proof that an associative Latin square necessarily forms a group, clarifying the algebraic structure underlying Latin squares.

Findings

Associative Latin squares are groups

The proof is notably short and elegant

Connects combinatorics with algebra

Abstract

A Latin square of order $n$ with symbols $a_1,\ldots,a_n$ can be considered as a multiplication table for binary operation in the set $A=\{a_1,\ldots,a_n\}$. We prove that, if this operation is associative, then $A$ is a group.