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

From Groupprops

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

## Contents

## Definition

A **group of nilpotency class two whose commutator map is the double of a skew-symmetric 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 .
- for all .

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

## Examples

- Any LCS-Baer Lie group provides an example. In the finite case, this means that any group of nilpotency class two whose 2-Sylow subgroup is abelian gives an example.
- Other than these, the smallest finite examples are M16 and central product of D8 and Z4, both groups of order 16.

## Relation with other properties

### Stronger properties

### 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 |