Topological closure of subgroup belonging to a subvariety of the variety of groups is in the same subvariety

Statement

In terms of varieties

Suppose $G$ is a topological group and $H$ is a subgroup that belongs to a subvariety $\mathcal{V}$ of the variety of groups. Then, the closure of $H$ in $G$ is a closed subgroup of $G$ that also belongs to $\mathcal{V}$.

In terms of group property

Suppose $G$ is a topological group and $H$ is a subgroup that satisfies a varietal group property (or quasivarietal group property) $p$ of the variety of groups. Then, the closure of $H$ in $G$ is a closed subgroup of $G$ that also satisfies $p$.

Particular cases

varietal group property	corresponding statement
abelian group	topological closure of abelian subgroup is abelian
nilpotent group of nilpotency class $c$	topological closure of nilpotent group is nilpotent of same class
solvable group of derived length $\ell$	topological closure of solvable group is solvable of same derived length

Related facts

Similar facts

• Topological closure of subgroup is subgroup

Opposite facts

• Topological closure of cyclic subgroup need not be cyclic (in fact, such a closure is termed a topologically cyclic group)
• [[Topological closure of finitely generated subgroup need not be finitely generated (in fact, such a closure is termed a topologically finitely generated group)