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

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 an alternating bihomomorphism giving class two** is a group satisfying **both** the following:

- is a group of nilpotency class two.
- There is a function such that for all , where denotes the commutator in the group, is a bihomomorphism, is the identity element and for all and is the identity element for all .

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}} |