# Normality satisfies lower central series condition

From Groupprops

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)

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## Statement

Suppose is a group and is a normal subgroup of . Suppose is a positive integer. Denote by the member of the lower central series of , and denote by the member of the lower central series of . Then, is a normal subgroup of .

## Facts used

- Lower central series member functions are monotone, i.e., if , then .
- Normality is preserved under any monotone subgroup-defining function

## Proof

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