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