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

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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

Stronger properties

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