# Lower central series members are divisibility-closed in nilpotent group

From Groupprops

## Contents

## Statement

Suppose is a nilpotent group. Then, all members of the lower central series of are divisibility-closed subgroups of .

## Related facts

### Similar facts

- Derived subgroup is divisibility-closed in nilpotent group
- Derived series members are divisibility-closed in nilpotent group

### Opposite facts

### Applications

## Facts used

- Equivalence of definitions of nilpotent group that is divisible for a set of primes: We are interested in the (1) implies (4) part of the equivalence for the "nilpotent group and a prime" case: if is a -divisible nilpotent group, then each of the quotients is also -divisible for positive integers .

## Proof

The proof follows from the (1) implies (4) implication of Fact (1). Specifically, if we want to show that is divisibility-closed in , we make two cases:

- has class less than : In this case, is trivial, and there is nothing to prove.
- has class or higher: In this case, if is the class of , set .