Uniformly fully inert subgroups of abelian p-groups

A condition on abelian p-groups, that of being finitely fully transitive, is defined and studied. It is shown that many groups satisfy this condition, including all of the separable groups. We also consider the class of groups G for which there is an ordinal \(\delta \) $\delta$ such that \(p^\delta G\) $p^{\delta }G$ is finite and \(G/p^\delta G\) $G/p^{\delta }G$ is universally fully transitive. It is shown that the groups in either of these classes have the property that every uniformly fully inert subgroup is commensurable with a fully invariant subgroup. This gives a positive answer to a conjecture of Dardano et al. (Int J Group Theory 7(3):17–62, 2018) for the groups in these classes.