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

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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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A **group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle** is a group satisfying the following condition: there is a function satisfying the following conditions:

- for all .
- for all .
- is the identity element whenever is cyclic.
- is the identity element for all .
- for all .

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 |