On relations between generalized norms in locally finite groups

Abstract

In the paper the relations between such generalized norms as the norm of Abelian non-cyclic subgroups and the norm of decomposable subgroups in the class of infinite locally finite groups are studied. The local nilpotency and non-Dedekindness of the norm of Abelian non-cyclic subgroups are considered as the restrictions. It was proved that any infinite locally finite group with mentioned restrictions on the norm of Abelian non-cyclic subgroups is a finite extension of a quasicyclic \(p\)-subgroup and does not contain Abelian non-cyclic \(p'\)-subgroups. Moreover, in such groups the norm of Abelian non-cyclic subgroups necessarily includes Abelian non-cyclic subgroups and therefore is a non-Hamiltonian \(\overline{HA}\)-group (i.e., a group with the normality condition for Abelian non-cyclic subgroups), whose structure is known. It was shown that for infinite locally finite groups with the non-Dedekind locally nilpotent norm \(N_G^A\) the relation \(N^A_G \supseteq N^d_G\) holds. The inclusion is proper for infinite torsion non-primary locally nilpotent groups with the mentioned restrictions on the norm \(N_G^A\), as well as for infinite locally finite groups in which the norm \(N_G^A\) is a non-Dedekind non-primary locally nilpotent group.