latin square

Hall-Paige conjecture ★★★

Author(s): Hall; Paige

A \emph{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.

\begin{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. \end{conjecture}

Keywords: complete map; finite group; latin square

Snevily's conjecture ★★★

Author(s): Snevily

\begin{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. \end{conjecture}

Keywords: addition table; latin square; transversal

Even vs. odd latin squares ★★★

Author(s): Alon; Tarsi

A \Def{latin square} is \emph{even} if the product of the signs of all of the row and column permutations is 1 and is \emph{odd} otherwise.

\begin{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. \end{conjecture}

Keywords: latin square

Rota's basis conjecture ★★★

Author(s): Rota

\begin{conjecture} Let $V$ be a vector space of dimension $n$ and let $B_1,\ldots,B_n \subseteq V$ be bases. Then there exist $n$ disjoint transversals of $B_1,\ldots,B_n$ each of which is a base. \end{conjecture}

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

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