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Latin Squares

1) Show that for n less than or equal to 4, any Latin square of order n can be obtained from the multiplication table of a group by permuting rows, columns, and symbols. Show that this is not true for n=5

2) If n is an order for which mutually orthogonal Latin squares exist, does every Latin square of order n have an orthogonal partner?

Solution Preview

1) There is a unique group of each order 1, 2, 3, and two of order 4 (up to iso-morphism); check that all these give the Latin squares.
For n = 5, the cyclic group is the only type of group , and gives a Latin square
containing no subsquare of order 2. ...

Solution Summary

Latin squares are investigated.