Orbit groups

TL;DR

This paper investigates two orbit-generated groups from a group action, analyzing their properties and a natural homomorphism, revealing it is neither injective nor surjective, and describing its kernel.

Contribution

It introduces and studies properties of two orbit groups related to group actions and homology, providing new insights into their structure and the nature of the associated homomorphism.

Findings

The homomorphism between the orbit groups is not injective.

The homomorphism between the orbit groups is not surjective.

A description of the kernel of the homomorphism is provided.

Abstract

In the paper "Aquino, C., Jim\'enez, R., Mijangos, M., Morales Mel\'endez, Q.: On Invariant (co)homology of a group, preprint" are introduced two groups generated by the orbits of an action of a group on another group by automorphisms. One is of group-theoretic nature and the other comes from homology of invariant group chains. In this note are given some properties of the first groups and is studied a natural homomorphism between these groups. More precisely, it is shown that this homomorphism is not injective nor surjective. A description of the kernel is given. Note: there was a previous version with an error in the construction. The error has been corrected now.