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

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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

• Any LCS-Baer Lie group is an example. In particular, this means that any group of nilpotency class two whose 2-Sylow subgroup is abelian gives an example.
• The smallest example that is not a LCS-Baer Lie group is SmallGroup(64,57), a nilpotent group of order 64.

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