Subgroup transitivity in abelian groups

Abstract:

We generalize an appropriate modification of the classical notion of transitivity in abelian $p$-groups from one that is based on elements to one based on subgroups. We consider those $p$-groups that are transitive in the sense that there is an automorphism of the group that maps one isotype subgroup $H$ onto any other isotype subgroup $H’$, unless this is impossible due to the simple reason that either the subgroups are not isomorphic or the quotient groups are not (as valuated groups when endowed with the coset valuation). Slight variations of this are used to define the classes of strongly transitive and strongly U-transitive groups. The latter class is studied in some detail in this paper, and it is shown that every $C_\Omega$-group is strongly transitive with respect to countable isotype subgroups.