Normality satisfies lower central series condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., lower central series condition)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal subgroup |Get facts that use property satisfaction of normal subgroup | Get facts that use property satisfaction of normal subgroup|Get more facts about lower central series condition

Statement

Suppose $G$ is a group and $H$ is a normal subgroup of $G$. Suppose $k$ is a positive integer. Denote by ${\gamma }_k(G)$ the $k^{th}$ member of the lower central series of $G$, and denote by ${\gamma }_k(H)$ the $k^{th}$ member of the lower central series of $H$. Then, ${\gamma }_k(H)$ is a normal subgroup of ${\gamma }_k(G)$.

Facts used

1. Lower central series member functions are monotone, i.e., if $H\le G$, then ${\gamma }_k(H)\le {\gamma }_k(G)$.
2. Normality is preserved under any monotone subgroup-defining function

Proof

The proof follows directly by combining Facts (1) and (2).