Pseudovarietal group property

Definition
A group property $$\alpha$$ is termed pseudovarietal if it satisfies the following three conditions:


 * 1) It is a subgroup-closed group property, i.e., whenever $$G$$ is a group satisfying $$\alpha$$ and $$H$$ is a subgroup of $$G$$, $$H$$ also satisfies $$\alpha$$.
 * 2) It is a quotient-closed group property, i.e., whenever $$G$$ is a group satisfying $$\alpha$$ and $$H$$ is a normal subgroup of $$G$$, the quotient group $$G/H$$ also satisfies $$\alpha$$.
 * 3) It is a finite direct product-closed group property, i.e., whenever $$G_1,G_2,\dots,G_n$$ are groups all of which satisfy $$\alpha$$, the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ also satisfies $$\alpha$$.