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.

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 $$\gamma_k(G)$$.