Extensions of Abelian Groups of Finite Rank

Abstract

Every abelian group $X$ of finite rank arises as the middle group of an extension $e:0 \to F \to X \to T \to 0$ where $F$ is free of finite rank $n$ and $T$ is torsion with the $p$-ranks of $T$ finite for all primes $p$. Given such a $T$ and $F$ we study the equivalence classes of such extensions which result from stipulating that two extensions ${e_i}:0 \to F \to {X_i} \to T \to 0,i = 1,2$, are equivalent if ${e_1} = \beta {e_2}\alpha$ for $\alpha \in \operatorname {Aut} (T)$ and $\beta \in \operatorname {Aut} (F)$. We reduce the problem to $T$ $p$-primary of finite rank, where in the one case $T$ is injective, and in the other case $T$ is reduced. Suppose $T = \Pi _{i = 1}^m{T_i}$. In our main theorems we prove that in each case these equivalence classes of extensions are in 1-1 correspondence with the equivalence classes of $n$-generated subgroups of $E$ where $E = \Pi _{i = 1}^m{E_i},{E_i} = \operatorname {End} ({T_i})$. Two $n$-generated subgroups of $E$ will be called equivalent if one can be mapped onto the other by an automorphism of $E$.