Lower central series member functions are monotone

Statement

Each of the subgroup-defining functions corresponding to members of the lower central series is a monotone subgroup-defining function.

Suppose $H$ is a subgroup of a group $G$. Then, for any positive integer $k$, $\gamma _{k}(H)$ is a subgroup of $\gamma _{k}(G)$, where $\gamma _{k}$ denotes the $k^{th}$ member of the lower central series.

Related facts

Applications

• Normality is preserved under any monotone subgroup-defining function

Proof

Given: A group $G$, a subgroup $H$ of $G$. Denote by $\gamma _{k}(G)$ and $\gamma _{k}(H)$ respectively the $k^{th}$ members of the lower central series of $G$ and $H$.

To prove: $\gamma _{k}(H)$ is a subgroup of $\gamma _{k}(G)$.

Proof: This follows directly from the definition. Recall that:

$\gamma _{k}(H)=\langle [[\dots [h_{1},h_{2}],\dots ,h_{k-1}],h_{k}]\mid h_{1},h_{2},\dots ,h_{k}\in H\rangle$

and

$\gamma _{k}(G)=\langle [[\dots [g_{1},g_{2}],\dots ,g_{k-1}],g_{k}]\mid g_{1},g_{2},\dots ,g_{k}\in G\rangle$

Note that any $k$-fold iterated left-normed commutator in $H$ is also a $k$-fold iterated left-normed commutator in $G$. Hence, the generating set for $\gamma _{k}(H)$ is a subset of the generating set for $\gamma _{k}(G)$. Thus, $\gamma _{k}(H)$ is a subgroup of $\gamma _{k}(G)$.