On the existence of normal complement to the Sylow subgroups of some infinite groups

W. Burnside has proved that if a Sylow p–subgroup P of a finite group G is abelian and NG(P) = CG(P), then P has a normal complement, that is G is p–nilpotent. This result has been extended by A. Ballester-Bolinches and R. Esteban-Romero that have shown that if a Sylow p–subgroup P of a finite group G is modular and NG(P) = P CG(P), then G is p–nilpotent. In this paper we generalize the latter result to infinite groups. We show that a hyperfinite group G with a Sylow p–subgroup S that is modular and pronormal is p–nilpotent if and only if NG(S) is p-nilpotent.