This paper is concerned with
transitivity and full transitivity of primary abelian groups. It is well known
that countable primary groups and primary groups without elements of
infinite height are both transitive and fully transitive. The question of whether
all primary groups are transitive or fully transitive was recently answered
negatively by C. Megibben. Megibben’s examples indicate that pωG may be
transitive (fully transitive) while G is not transitive (fully transitive). For β an
ordinal number, we investigate conditions on a primary group G which will
insure that G is transitive (fully transitive) whenever pβG is transitive (fully
transitive). Specifically, we show that if G∕pβG is a direct sum of countable
groups and pβG is fully transitive, then G is fully transitive. The same result
is established for transitivity except that β is restricted to be a countable
ordinal.
