# latin square

## Hall-Paige conjecture ★★★

Author(s): Hall; Paige

A complete map for a (multiplicative) group is a bijection so that the map is also a bijection.

Conjecture   If is a finite group and the Sylow 2-subgroups of are either trivial or non-cyclic, then has a complete map.

Keywords: complete map; finite group; latin square

## Snevily's conjecture ★★★

Author(s): Snevily

Conjecture   Let be an abelian group of odd order and let satisfy . Then the elements of and may be ordered and so that the sums are pairwise distinct.

Keywords: addition table; latin square; transversal

## Even vs. odd latin squares ★★★

Author(s): Alon; Tarsi

A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise.

Conjecture   For every positive even integer , the number of even latin squares of order and the number of odd latin squares of order are different.

Keywords: latin square

## Rota's basis conjecture ★★★

Author(s): Rota

Conjecture   Let be a vector space of dimension and let be bases. Then there exist disjoint transversals of each of which is a base.

Keywords: base; latin square; linear algebra; matroid; transversal