On a Generalization of the Notion of Semidirect Product of Groups

TL;DR

This paper generalizes the concept of semidirect products of groups by introducing an external version, providing algebraic criteria and algorithms for their construction and homomorphisms, with applications to hypercrossed complexes.

Contribution

It introduces an external generalization of the internal r-fold semidirect product of groups, along with algorithms and criteria for their algebraic properties and homomorphisms.

Findings

Provides an algorithmic procedure for axioms of external SDP

Establishes criteria for homomorphisms from factors to entire SDP

Lays groundwork for algebraic axioms of hypercrossed complexes

Abstract

We introduce an external version of the internal r-fold semidirect product of groups (SDP) of Carrasco and Cegarra. Just as for the classical external SDP, certain algebraic data are required to guarantee associativity of the construction. We give an algorithmic procedure for computing axioms characterizing these data. Additionally, we give criteria for determining when a family of homomorphisms from the factors of an SDP into a monoid or group assemble into a homomorphism on the entire SDP. These tools will be used elsewhere to give explicit algebraic axioms for hypercrossed complexes, which are algebraic models for classical homotopy types introduced by Carrasco and Cegarra.