# Normal complex of groups

## Definition

A chain complex of groups $(G_n,d_n)_{n \in \mathbb{Z}}$ given as follows:

$\dots \to G_n \stackrel{d_n}{\to} G_{n-1} \stackrel{d_{n-1}}{\to} G_{n-2} \to \dots$

is termed a normal complex of groups if, for every $n \in \mathbb{Z}$, the image group $d_n(G_n)$ is a normal subgroup inside $G_{n-1}$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
chain complex of abelian groups all the groups are abelian follows from abelian implies every subgroup is normal |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions