On two problems concerning extension groups of Abelian groups

P. A. Krylov

Sbornik. Mathematics, Tome 81 (1995) no. 1, pp. 59-76

Résumé

This paper deals with the extension group $\operatorname{Ext}(A,C)$ of an abelian group $C$ by an abelian group $A$. In § 1 the problem of how the groups $A$, $B$ are related to one another if $\operatorname{Ext}(A,C)\cong\operatorname{Ext}(B,C)$ for any group $C$ is completely solved for a torsion-free group $A$ of finite rank (Theorem 1.7). Also studied are conditions under which the group $\operatorname{Ext}(A,B)$ is torsion-free. Theorem 2.5 describes the torsion-free groups $A$, $B$ of finite rank with the property, more general than the situation in [13], that both $\operatorname{Ext}(A,B)$ and $\operatorname{Ext}(B,A)$ are torsion-free.