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 $G$.

Related facts

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. 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 $i=m,j=c+1$.