# Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle

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## Definition

A group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle is a group $G$ satisfying the following condition: there is a function $\circ:G \times G \to G$ satisfying the following conditions:

• $x \circ y \in Z(G)$ for all $x,y \in G$.
• $(x \circ (yz))(y \circ z) = ((xy) \circ z)(x \circ y)$ for all $x,y,z \in G$.
• $x \circ y$ is the identity element whenever $\langle x,y \rangle$ is cyclic.
• $x \circ (y \circ z)$ is the identity element for all $x,y,z \in G$.
• $x \circ y = (y \circ x)^{-1}$ for all $x,y \in G$.
• $xyx^{-1}y^{-1} = (x \circ y)^2$ for all $x,y \in G$.

This is precisely the kind of group that can participate in the cocycle halving generalization of Baer correspondence.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group any two elements commute CS-Baer Lie group, 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 class at most two, and 2-powered (every element has a unique square root) CS-Baer Lie group, 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 class at most two, and derived subgroup is 2-powered CS-Baer Lie group, 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 class at most two, and center is 2-powered CS-Baer Lie group, LUCS-Baer Lie group|FULL LIST, MORE INFO
LUCS-Baer Lie group class at most two, and derived subgroup has unique square roots in center CS-Baer Lie group|FULL LIST, MORE INFO
CS-Baer Lie group class at most two, and derived subgroup has unique square roots in an intermediate subgroup within the center 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 the commutator map has a linear "half" |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle |FULL LIST, MORE INFO
group whose derived subgroup is contained in the square of its center every element of the derived subgroup has a square root in the center |FULL LIST, MORE INFO
group that is 1-isomorphic to an abelian group Group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle|FULL LIST, MORE INFO
group of nilpotency class two Group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle, Group whose derived subgroup is contained in the square of its center|FULL LIST, MORE INFO