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

## Statement

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

## Facts used

1. 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 $G$ is a $p$-divisible nilpotent group, then each of the quotients $\gamma_i(G)/\gamma_j(G)$ is also $p$-divisible for positive integers $i < j$.

## Proof

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

• $G$ has class less than $m$: In this case, $\gamma_m(G)$ is trivial, and there is nothing to prove.
• $G$ has class $m$ or higher: In this case, if $c$ is the class of $G$, set $i = m, j = c + 1$.