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

## Contents

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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 skew of a 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 $x,y,z \in G$.
• $xyx^{-1}y^{-1} = (x \circ y)(y \circ x)^{-1}$ for all $x,y \in G$.

This is precisely the kind of group that can participate in the cocycle skew reversal 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
Baer Lie group
LCS-Baer Lie group
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

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group 1-isomorphic to an abelian group
group of nilpotency class two