Generalized Dieudonné and Hill criteria

Abstract

Let $A$ be a reduced abelian $p$-group with a nice subgroup $G$. It is proved that if $A/G$ is simply presented, then $A$ is simply presented precisely when $G$ is strongly simply presented in $A$. Moreover, the same type theorem for the class of ${\aleph }_1$-$\Sigma$-cyclic $p$-groups is also established without the niceness of $G$ in $A$. Some analogous assertions for other exotic sorts of abelian groups are also considered.

The results obtained strengthen previous results due to J. A. Dieudonné (Portugal. Math., 1952), P. D. Hill (Trends in Math., 1999), and some other authors.