The pronorm of a group G is the set $P(G)$ of all elements $g\in G$ such that X and $X^g$ are conjugate in ${\langle {X,X^g}\rangle }$ for every subgroup X of G. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble group G whose pronorm contains a subgroup of finite index.

A group $G$ is said to have the $T$ -property (or to be a $T$ -group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality  $\aleph$ whose proper subgroups of cardinality $\aleph$ have a transitive normality relation. It is proved that such a group  $G$ is a $T$ -group (and all its subgroups have the same property) provided that $G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if $G$ is an uncountable soluble group of cardinality $\aleph$ whose proper normal subgroups of cardinality  $\aleph$ have the $T$ -property, then every subnormal subgroup of $G$ has only finitely many conjugates.

We extend to soluble $\text{FC}^{\ast }$-groups, the class of generalised FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J. 28(3) (2002), 241–254], the characterisation of finite soluble T-groups obtained recently in Kaplan [‘On T-groups, supersolvable groups, and maximal subgroups’, Arch. Math. (Basel) 96(1) (2011), 19–25].