Coxeter–Chein Loops

Abstract

In 1974, Orin Chein discovered a new family of Moufang loops which are now called Chein loops. Such a loop can be created from any group W together with ℤ₂ by a variation on a semidirect product. We first settle an open problem, originally proposed by Petr Vojtěchovský in 2003, by finding a minimal presentation for the Chein loop with respect to a presentation for W. We then study these loops in the case where W is a Coxeter group and show that it has what we call a Chein-Coxeter system, a small set of generators of order 2, together with a set of relations closely related to the Coxeter relations and Chein relations. In particular, even if the Moufang loop is infinite, it is finitely presented. Viewing these presentations as amalgams of loops, we then apply methods due to Blok and Hoffman to describe a family of twisted Coxeter–Chein loops.

Key Words:

• Amalgams
• Coxeter groups
• Generators and relations
• Moufang loops

2010 Mathematics Subject Classification:

• 20F55
• 20F05
• 20N05
• 18G30

Notes

Communicated by I. Shestakov.