The pro-norm of a profinite group

In “classical” group theory, a subgroup which is often relevant in the study of a group G is the norm, i.e., the intersection of all normalizers of subgroups of G. The main aim of this paper is to introduce the analogous concept of pro-norm for a profinite group and to investigate its relation to the norm. In order to understand this connection, we first investigate profinite groups whose closed proper subgroups are normal or abelian. This will also naturally lead to the concept of pro-metanorm, which, in turns, generalize another very useful characteristic subgroup of an arbitrary group. Finally, other restrictions on closed proper subgroups of a profinite group are investigated.