AbstractThenormof 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 embedding properties of themetanormof a group, defined as the intersection of all normalizers of non-abelian subgroups. The metanorm is related to the so-calledmetahamiltonian groups, i.e. groups in which all non-abelian subgroups are normal, and it is known that every locally graded metahamiltonian group is finite over its second centre. Among other results, it is proved here that ifGis a locally graded group whose metanormMis not nilpotent, then{M^{\prime}/M^{\prime\prime}}is a small eccentric chief factor and it is the only obstruction to a strong hypercentral embedding ofMinG.
