Generalised norms in finite soluble groups

We give a framework for a number of generalisations of Baerʼs norm that have appeared recently. For a class CC of finite nilpotent groups we define the CC-norm κC(G)κC(G) of a finite group G to be the intersection of the normalisers of the subgroups of G   that are not in CC. We show that those groups for which the CC-norm is not hypercentral have a very restricted structure. The non-nilpotent groups G   for which G=κC(G)G=κC(G) have been classified for some classes. We give a classification for nilpotent classes closed under subgroups, quotients and direct products of groups of coprime order and show the known classifications can be deduced from our classification.