Equalizers and coactions of groups
Volume 171 / 2002
Fundamenta Mathematicae 171 (2002), 155-165
MSC: Primary 20E06; Secondary 20F99.
DOI: 10.4064/fm171-2-3
Abstract
If $f:G\to H$ is a group homomorphism and $p,q$ are the projections from the free product $G*H$ onto its factors $G$ and $H$ respectively, let the group ${\cal E}_f\subseteq G*H$ be the equalizer of $fp$ and $q:G*H\to H$. Then $p$ restricts to an epimorphism $p_f=p|{\cal E}_f:{\cal E}_f\to G$. A right inverse (section) $G\to {\cal E}_f$ of $p_f$ is called a coaction on $G$. In this paper we study ${\cal E}_f$ and the sections of $p_f$. We consider the following topics: the structure of ${\cal E}_f$ as a free product, the restrictions on $G$ resulting from the existence of a coaction, maps of coactions and the resulting category of groups with a coaction and associativity of coactions.
