Peiffer commutators by using GAP package (Q1273279)

Simplicial groups provide algebraic models for all connected homotopy types. The semidirect product decompositions of each level of a simplicial group provide useful information on the structure of the corresponding homotopy type. These decompositions naturally give Peiffer commutators by using the commutators of lower dimensional elements then reflecting the result into the Moore complex that is at the core of the decomposition. The Peiffer pairings \(F_{\alpha,\beta}\) that result provide a pairing hypercrossed complex structure on the Moore complex of the simplicial group. In this paper the author briefly discusses the higher dimensional Peiffer commutators, \(F_{\alpha,\beta}\) and lists a GAP program that will produce the formulae for the \(F_{\alpha,\beta}\).