# Divisibility-closedness is strongly join-closed in nilpotent group

From Groupprops

This article gives the statement and possibly, proof, of a subgroup property satisfying a subgroup metaproperty, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), the subgroup property (i.e., Divisibility-closed subgroup (?)) satisfies the metaproperty (i.e., Strongly join-closed subgroup property (?))

View all subgroup metaproperty satisfactions in nilpotent groups View all subgroup metaproperty satisfactions in nilpotent groups View all subgroup metaproperty satisfactions View all subgroup metaproperty dissatisfactions

## Contents

## Statement

Suppose is a nilpotent group and are all divisibility-closed subgroups of . Then, the join of subgroups is also a divisibility-closed subgroup of .

## Related facts

- Powering-invariance is strongly join-closed in nilpotent group
- Divisibility-closedness is not finite-join-closed (the examples for this are solvable, but cannot be nilpotent)
- Divisibility-closedness is not finite-intersection-closed (there is an abelian example)
- Powering-invariance is strongly intersection-closed

## Facts used

## Proof

The proof follows from Fact (1): simply take the set-theoretic union of the subgroups as the "divisible subset" for the appropriate set of primes and argue that the subgroup generated by it is also appropriately divisible.