Finite extensions of A-solvable abelian groups

An abelian group G is almost A-solvable if the natural map θG: Hom(A,G)circle times operatorE(A) A→G is a quasi-isomorphism. Two strongly indecomposable torsion-free abelian groups A and B of finite rank are quasi-isomorphic if and only if the classes of almost A-solvable and almost B-solvable groups coincide. Homological properties of almost A-solvable groups are described, and several examples are given. In particular, there exists a torsion-free almost A-solvable group which is not quasi-isomorphic to an A-solvable group.