Residually finite extensions of periodic groups (Q1204445)

The object of this paper is to establish the existence of a residually finite group \(A\) and a homomorphism \(\theta\) defined on \(A\) with some useful properties. First of all, a copy of any countable group is embedded in \(A\theta\). Second, when \(\theta\) is restricted to certain subgroups of \(A\) to be defined in the sequel, the kernel of \(\theta\) is periodic. The group \(A\) arises in a natural way as a group of automorphisms. Our methods enable us to establish: Theorem A. Given any finitely generated group \(T\) generated by elements of finite order, we may find a finitely generated residually finite group \(B = B_ T\) generated by elements of finite order and a periodic normal subgroup \(K = K_ T\) of \(B\) with \(B/K \approx T\). If \(T\) is finitely presented, \(K\) can be taken to be the normal closure of a finite number of elements of \(B\).