The equivalence of high subgroups

Abstract:

Two subgroups of a group $G$ are called equivalent if there is an automorphism of $G$ that maps one of the subgroups onto the other. Suppose that $G$ is a $p$-primary abelian group and that $\lambda$ is an ordinal. A subgroup $H$ of $G$ is ${p^\lambda }$-high in $G$ if $H$ is maximal in $G$ with respect to having zero intersection with ${p^\lambda }G$. Under certain conditions on the quotient group $G/{p^\lambda }G$ slightly weaker than total projectivity, it is shown, for a given $\lambda$, that any two ${p^\lambda }$-high subgroups of $G$ are equivalent. In particular, if $G/{p^\omega }G$ is ${p^{\omega + 1}}$-projective, the ${p^\omega }$-high subgroups of $G$ are all equivalent.