# Commutator-verbal implies divisibility-closed in nilpotent group

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Commutator-verbal subgroup (?)) must also satisfy the second subgroup property (i.e., Divisibility-closed subgroup (?)). In other words, every commutator-verbal subgroup of nilpotent group is a divisibility-closed subgroup of nilpotent group.

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## Statement

Suppose is a nilpotent group and is a commutator-verbal subgroup of . Then, is a divisibility-closed subgroup of . Explicitly, for any prime number such that is a -divisible, is also -divisible.

In particular, this also shows that is a powering-invariant subgroup of .