A random Hall-Paige conjecture

Abstract A complete mapping of a group $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is a bijection $\phi \colon G\to G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>G</mml:mi> </mml:math> such that $x\mapsto x\phi (x)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>↦</mml:mo> <mml:mi>x</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> has a complete mapping whenever $\prod _{x\in G} x$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>∏</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mi>x</mml:mi> </mml:math> is the identity in the abelianization of $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . This was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. In this paper, we give a combinatorial proof of a far-reaching generalisation of the Hall-Paige conjecture for large groups. We show that for random-like and equal-sized subsets $A$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> , $B$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> , $C$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> of a group $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , there exists a bijection $\phi \colon A\to B$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>B</mml:mi> </mml:math> such that $x\mapsto x\phi (x)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>↦</mml:mo> <mml:mi>x</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> is a bijection from $A$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> to $C$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> whenever $\prod _{a\in A} a \prod _{b\in B} b=\prod _{c\in C} c$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>∏</mml:mo> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> <mml:mi>a</mml:mi> <mml:msub> <mml:mo>∏</mml:mo> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∏</mml:mo> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:msub> <mml:mi>c</mml:mi> </mml:math> in the abelianization of $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . We use this statement as a black-box to settle the following old problems in combinatorial group theory for large groups. (1) We characterise sequenceable groups, that is, groups which admit a permutation $\pi $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> of their elements such that the partial products $\pi _{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , $\pi _{1}\pi _{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , $\pi _{1}\pi _{2}\cdots \pi _{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⋯</mml:mo> <mml:msub> <mml:mi>π</mml:mi>