Co-actions of groups (Q2719730)

Let \(f\colon G\to H\) be a fixed homomorphism and \(p'\colon G*H\to G\) and \(p''\colon G*H\to H\) the two projections of the free product. Then a co-action relative to \(f\) is a homomorphism \(s\colon G\to G*H\) such that \(p's=id\) and \(p''s=f\) where \(id\) is the identity homomorphism. The authors study this notion.NEWLINENEWLINENEWLINEThey show a co-action \(s\colon G\to G*H\) relative to \(f\) gives a free product decomposition of \(G\) into a free subgroup and a free product of subgroups of \(\ker f\). The authors obtain a complete description of \(s\) on all finite subgroups of \(G\). They find a canonical set of generators of \(G\) in the case when the co-action \(s\) is associative. Also, the authors investigate the action of \(\Hom(H,B)\) on \(\Hom(G,B)\) induced by \(s\colon G\to G*H\).NEWLINENEWLINENEWLINELet \(m\colon H\to H*H\) be a homomorphism such that \(p'm=p''m=id\), where \(p',p''\colon H*H\to H\) are the two projections. Suppose \(A\) is subgroup of \(H\). It is proved if \(m(A)\subseteq A*H\) then \(A\) is a free factor of \(H\). In addition, the authors give several diverse examples of co-actions.