Serre class of abelian groups

Definition
Suppose $$\alpha$$ is a group property (satisfied only by (some) abelian groups) and let $$C$$ be the class of all abelian groups satisfying $$\alpha$$. We say that $$C$$ is a Serre class (or $$\alpha$$ is a Serre property) if it satisfies the following conditions:


 * 1) $$\alpha$$ is a subgroup-closed group property, or equivalently, the class $$C$$ is closed under taking subgroups.
 * 2) $$\alpha$$ is a quotient-closed group property, or equivalently, the class $$C$$ is closed under taking quotient groups.
 * 3) The class $$C$$ is closed under taking the tensor product of abelian groups.
 * 4) The class $$C$$ is closed under taking extensions where the extension group is itself abelian.
 * 5) The $$\operatorname{Tor}$$ group between two groups in the class $$C$$ is also in the class $$C$$.