Maria De Falco, Francesco de Giovanni, Leonid A. Kurdachenko, Carmela Musella
June 13, 2018

### Abstract

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 embedding properties of the metanorm of a group, defined as the intersection of all normalizers of non-abelian subgroups. The metanorm is related to the so-called metahamiltonian 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 if G is a locally graded group whose metanorm M is not nilpotent, then M′/M′′{M^{\prime}/M^{\prime\prime}} is a small eccentric chief factor and it is the only obstruction to a strong hypercentral embedding of M in G .