Quasiverbal subgroup

Definition
Suppose $$\mathcal{V}$$ is a subquasivariety of the variety of groups. Equivalently, the group property of being in $$\mathcal{V}$$ is a quasivarietal group property.

The $$\mathcal{V}$$-quasiverbal subgroup of a group $$G$$ is defined in the following equivalent ways:


 * It is the intersection of all normal subgroups $$N$$ of $$G$$ for which the quotient group $$G/N$$ is in $$\mathcal{V}$$.
 * It is the unique smallest normal subgroup of $$G$$ for which the quotient group is in the quasivariety $$\mathcal{V}$$.