# Latin square property

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.

References

*"Groups", C. R. Jordan and D. A. Jordan, Newnes, ISBN 0-340-61045-X

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Graeco-Latin square — Orthogonal Latin squares of order 3 Orthogonal Latin squares of order 5 In mathematics, a Graeco Latin square or Euler square or orthogonal Latin squares of order n over two …   Wikipedia

• Latin America, history of — Introduction       history of the region from the pre Columbian period and including colonization by the Spanish and Portuguese beginning in the 15th century, the 19th century wars of independence, and developments to the end of World War II.… …   Universalium

• Latin American art — Introduction       artistic traditions that developed in Mesoamerica, Central America, and South America after contact with the Spanish and Portuguese beginning in 1492 and 1500, respectively, and continuing to the present.       This article… …   Universalium

• Problems in Latin squares — In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for… …   Wikipedia

• Magic square — In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.[1] A normal magic… …   Wikipedia

• Madison Square Garden — Infobox Stadium stadium name=Madison Square Garden nickname=MSG, The Garden, The World s Most Famous Arena Landscape=yes location=4 Pennsylvania Plaza Manhattan, New York City, NY 10121 broke ground= opened=Former locations: 1879, 1890, 1925… …   Wikipedia

• Market Square (Lviv) — Coordinates: 49°50′29″N 24°01′53″E﻿ / ﻿49.841466°N 24.031265°E﻿ / 49.841466; 24.031265 …   Wikipedia

• Cluny Square Hotel Paris (Paris) — Cluny Square Hotel Paris country: France, city: Paris (Notre Dame Latin Quarter) Cluny Square Hotel Paris Conveniently located at the crossroads of the two famous boulevards, Saint Michel and Saint Germain, Cluny Square Hotel Paris offers its… …   International hotels

• Church of Saint Mary the Virgin (Times Square, New York) — St. Mary the Virgin Episcopal Church View of the church Denomination Episcopal Church Churchmanship Anglo Catholic Website …   Wikipedia

• Fitzroy Square — is one of the Georgian squares in London and is the only one found in the central London area known as in Fitzrovia.The square, nearby Fitzroy Street and the Fitzroy Tavern in Charlotte Street have the family name of Charles FitzRoy, 2nd Duke of… …   Wikipedia

### Share the article and excerpts

##### Direct link
Do a right-click on the link above
and select “Copy Link”