Abstract:

Coxeter groups, better known as reflection-generated groups, have numerous applications in various fields of mathematics and beyond. Groups with Fischer 3-transpositions are also associated with many structures: finite simple groups, triple graphs, geometries of various spaces, Lie algebras, etc.
In previous works, the authors established a simple genetic relationship between Coxeter groups and groups with symplectic 3-transpositions Fisher's ---symplectic and orthogonal groups over a field of two elements. As it turned out, Fisher groups are obtained from Coxeter groups using a single relation - the square of the product of two conjugate involutions, one of which belongs to the generating set of the Coxeter group, and the second is specially selected. Elements of computer calculations using the $GAP$ system were used. In this paper, the genetic codes of the commutators of these groups are found.  The series of Coxeter graphs used in the work are provided with markup indicating how