Published online by Cambridge University Press: 20 November 2018
The intersection IW(G) of the normalizers of the infinite subnormal subgroups of a group G is a characteristic subgroup containing the Wielandt subgroup W(G) which we call the generalized Wielandt subgroup. In this paper we show that if G is infinite, then the structure of IW(G)/ W(G) is quite restricted, being controlled by a certain characteristic subgroup S(G). If S(G) is finite, then so is IW(G)/ W(G), whereas if S(G) is an infinite Prüfer-by-finite group, then IW(G)/W(G) is metabelian. In all other cases, IW(G) = W(G).