# Group whose derived subgroup is contained in the square of its center

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

A group $G$ is termed group whose derived subgroup is contained in the square of its center if it satisfies the following equivalent conditions:

1. For every $x,y \in G$, there exists $z$ in the center $Z(G)$ such that $z^2 = [x,y]$, where $[x,y]$ denotes the commutator of $x$ and $y$.
2. The derived subgroup $G'$ is contained in the subgroup $\{z^2 \mid z \in Z(G) \}$ where $Z(G)$ is the center of $G$.

## Relation with other properties

### Weaker properties

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