Latin Square Property

In mathematics, the Latin square property is an elementary property of all groups. It states that if (G, *) is a group and a and b are elements of G, then there exists a unique element x of G such that a*x=b, and a unique element y of G such that y*a=b.

The Latin square property receives its name from the fact that for a finite group (G, *), it is possible (in principle) to draw a Cayley table, which gives the element a*b in the row corresponding to a and the column corresponding to b; the Latin Square property says that this table will be a Latin square, a square array in which each possible value for a cell appears precisely once in each row, and precisely once in each column. Further, for a countably infinite group G, it is possible to imagine an infinite array in which every row and every column corresponds to some element g of G, and where the element a*b is in the row corresponding to a and the column responding to b. In this situation too, the Latin Square property says that each row and each column of the infinite array will contain every possible value precisely once.

For an uncountably infinite group, such as the group of non-zero real numbers under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends. This is because the real numbers cannot all be written in a sequence, as they are uncountable.

For a proof of the first part of the Latin square property, see the Wikipedia page on elementary group theory (Theorem 1.3). The proof of the second part is similar.