On locally finite groups whose derived subgroup is locally nilpotent

Abstract

A celebrated theorem of Helmut Wielandt shows that the nilpotent residual of the subgroup generated by two subnormal subgroups of a finite group is the subgroup generated by the nilpotent residuals of the subgroups. This result has been extended to saturated formations in Ballester-Bolinches, Ezquerro, and Pedreza-Aguilera [Math. Nachr. 239–240 (2002), 5–10]. Although Wielandt's result is not true in arbitrary locally finite groups, we are able to extend it (even in a stronger form) to homomorphic images of periodic linear groups. Also, all results in Ballester-Bolinches, Ezquerro, and Pedreza-Aguilera [Math. Nachr. 239–240 (2002), 5–10] are extended to locally finite groups, so it is possible to characterize the class of locally finite groups with a locally nilpotent derived subgroup as the largest subgroup-closed saturated formation $X$ such that, for all $\mathrm{SL}$-closed saturated formations $F$, the $F$-residual of an $X$-group generated by $F$-subnormal subgroups is the subgroup generated by their $F$-residuals. Our proofs are based on a reduction theorem that is of an independent interest. Furthermore, we provide strengthened versions of Wielandt's result for other relevant classes of groups, among which we mention the class of paranilpotent groups. A brief discussion on the permutability of the residuals is given at the end of the paper.