The metanorm and its influence on the group structure

The norm of a group was introduced by R. Baer as the intersection of all normalizers of subgroups, and it was later proved that the norm is always contained in the second term of the upper central series of the group. The aim of this paper is to study the influence on a group G of the behaviour of its metanorm M(G), defined as the intersection of all normalizers of non-abelian subgroups of G. The metanorm is related to the so-called metahamiltonian groups, i.e. groups in which all non-abelian subgroups are normal, introduced by Romalis and Sesekin fifty years ago. Among other results, it is proved here that if G is any locally finite group whose metanorm is metabelian but not nilpotent, then G is metahamiltonian (or equivalently M(G)=G), unless the order of the commutator subgroup of M(G) is the square of a prime number.