On T-Extensions of Abelian Groups

Abstract

Let $Re$ be the category of all discrete abelian groups, and let $cal{L}$ be the category of all locally compact abelian (LCA) groups. For a group $Gin cal{L}$, the maximal torsion subgroup of $G$ is denoted by $tG$. A short exact sequence $0to Astackrel{phi}{to} Bstackrel{psi}{to}Cto 0$ in $Re$ is said to be a t-extension if $0to tAstackrel{phi}{to} tBstackrel{psi}{to}tCto 0$ is a short exact sequence. We show that the set of all t-extensions of $A$ by $C$ is a subgroup of $Ext(C,A)$, which contains $Pext(C,A)$ for discrete abelian groups $A$ and $C$. We establish conditions under which the t-extensions split and determine those groups in $Re$ which are t-injective or t-projective in $Re$. Finally we determine the compact groups $G$ in $cal{L}$ such that every pure extension of $G$ by a compact connected group $Cin cal{L}$ splits.