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

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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

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian group				|FULL LIST, MORE INFO
Baer Lie group				|FULL LIST, MORE INFO
LCS-Baer Lie group				|FULL LIST, MORE INFO
UCS-Baer Lie group				|FULL LIST, MORE INFO
LUCS-Baer Lie group				|FULL LIST, MORE INFO
CS-Baer Lie group				|FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two				|FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group of nilpotency class two				|FULL LIST, MORE INFO