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 $$G$$.

Similar facts

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

Opposite facts

 * Derived subgroup not is divisibility-closed

Applications

 * Lower central series members are quotient-powering-invariant in nilpotent group

Facts used

 * 1) uses::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 $$i = m, j = c + 1$$.