# Centralizer of divisibility-closed subgroup is completely divisibility-closed in nilpotent group

From Groupprops

## Statement

Suppose is a nilpotent group and is a divisibility-closed subgroup of . Then, the centralizer of in is a completely divisibility-closed subgroup of .

In particular, this shows that the property of being a divisibility-closed subgroup of nilpotent group is a centralizer-closed subgroup property, and also that the property of being a completely divisibility-closed subgroup of nilpotent group is a centralizer-closed subgroup property.

## Related facts

- Upper central series members are completely divisibility-closed in nilpotent group: In fact, the stated fact here can be viewed as a generalization of the fact that upper central series members are completely divisibility-closed.
- Kernel of a bihomomorphism implies completely divisibility-closed: The ideas used in the proof of the fact on the current page are a generalization and application of the kernel of a bihomomorphism idea.
- Kernel of a multihomomorphism implies completely divisibility-closed: The ideas used in the proof of the fact on the current page are a generalization and application of the kernel of a multihomomorphism idea.