Amalgamation of locally finite p-groups over a countable FC-group (Q1103051)

We develop modifications of the permutational product. These enable us to prove amalgamation theorems of a new type for locally finite groups over a common countable FC-subgroup. Our main conclusions are: Corollary 1. Let G and H be locally finite groups (locally finite p- groups) containing a common countable FC-subgroup U such that some normal series (chief series) \(\Sigma _ G\) in G resp. \(\Sigma _ H\) in H induce a common series in U consisting of U and finite normal subgroups. Let M resp. K be the intersection of all terms of \(\Sigma _ G\) resp. \(\Sigma _ H\) containing U. If \(| G:M|\) is countable and if \(H=K\), then the amalgam of G and H over U is contained in a locally finite group (locally finite p-group). Corollary 2. Let G and H be locally finite p-groups containing the Prüfer p-group \(U=C_{p^{\infty}}\). If \(| G:<U^ G>|\) is countable and if \(H=<U^ H>\), then the amalgam of G and H over U is contained in a locally finite p-group. Corollary 2 provides further evidence for the conjecture that \(C_{p^{\infty}}\) is an amalgamation basis in the class of all locally finite p-groups. As an application we show that every proper normal subgroup of an \(\omega _ 1\)-existentially complete locally finite p- group has uncountable index.