Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two

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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 of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two is a group G satisfying both the following:

  1. G is a group of nilpotency class two.
  2. There is a function \circ :  G \times G \to G such that (x \circ y)^2 = [x,y] for all x,y \in G, where [ , ] denotes the commutator in the group, \circ is a bihomomorphism, x \circ x is the identity element and x \circ y = (y \circ x)^{-1} for all x,y \in G and (x \circ y) \circ z is the identity element for all x,y,z \in G.

Note that for class two, the left and right conventions for commutator coincide, so it does not matter which one we pick.

This is precisely the type of group that can participate on the group side of the linear halving generalization of Baer correspondence.

Examples

Finite examples

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group the commutator map is trivial, and its half can also be taken to be trivial CS-Baer Lie group, LCS-Baer Lie group|FULL LIST, MORE INFO
Baer Lie group CS-Baer Lie group, LCS-Baer Lie group, LUCS-Baer Lie group, UCS-Baer Lie group|FULL LIST, MORE INFO
LCS-Baer Lie group CS-Baer Lie group|FULL LIST, MORE INFO
CS-Baer Lie group CS-Baer Lie group}}

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 double of a skew-symmetric cyclicity-preserving 2-cocycle |FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle|FULL LIST, MORE INFO
group whose derived subgroup is contained in the square of its center Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle|FULL LIST, MORE INFO
group that is 1-isomorphic to an abelian group the group is 1-isomorphic to an abelian 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 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 double of a skew-symmetric cyclicity-preserving 2-cocycle, 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
nilpotent group |FULL LIST, MORE INFO