latin square

Hall-Paige conjecture ★★★

Author(s): Hall; Paige

A complete map for a (multiplicative) group $ G $ is a bijection $ \phi : G \rightarrow G $ so that the map $ x \rightarrow x \phi (x) $ is also a bijection.

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

Keywords: complete map; finite group; latin square

Snevily's conjecture ★★★

Author(s): Snevily

Conjecture   Let $ G $ be an abelian group of odd order and let $ A,B \subseteq G $ satisfy $ |A| = |B| = k $. Then the elements of $ A $ and $ B $ may be ordered $ A = \{a_1,\ldots,a_k\} $ and $ B = \{b_1,\ldots,b_k\} $ so that the sums $ a_1+b_1, a_2+b_2 \ldots, a_k + b_k $ 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 $ n $, the number of even latin squares of order $ n $ and the number of odd latin squares of order $ n $ are different.

Keywords: latin square

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