Serre class of abelian groups

This article defines a group metaproperty: a property that can be evaluated to true/false for any group property
View a complete list of group metaproperties

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 $C$.