Cocycle skew reversal generalization of Baer correspondence
This is a generalization of the Baer correspondence (see also generalized Baer correspondence) between some Lie rings of class at most two and some groups of class at most two. Specifically, it is a correspondence:
In order to move back and forth between these structures, it is necessary to introduce an additional structure. This additional structure is that of a class two near-Lie cring. The additional structure choice is not unique; however, it turns out that different possible choices of the additional structure give rise to different ways of going back and forth but the group corresponding to a Lie ring remains the same up to isomorphism and vice versa.
From Lie ring to group
Suppose is a Lie ring arising as the skew of a class two near-Lie cring, i.e., there exists a binary operation such that:
- for all , i.e., is a 2-cocycle for trivial group action of on itself.
- if is cyclic.
- for all .
- for all .
Then, we can define a group structure on in terms of as follows:
|Group operation||Denoted as ...||Definition in terms of Lie ring operations|
It turns out that the group commutator is the same as the Lie bracket with these operations.
From group to Lie ring
Suppose is a group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle, i.e., there exists a function such that:
- for all .
- for all .
- is the identity element whenever is cyclic.
- is the identity element for all .
- for all .
Then we can define a Lie ring structure on as follows:
|Lie ring operation||Denoted as ...||Definition in terms of group operations|
|Lie bracket||, i.e., same as the commutator in the group.|
|Zero||Same as the group's identity element.|
Relation with other correspondences
|Name of correspondence||Eligible Lie rings||Eligible groups||Intermediate structure with extra information||Key idea of generalization||Rough description of scope||Group(s) of smallest order covered at this level of generality but at no preceding level||Order of these groups|
|Baer correspondence||Baer Lie rings||Baer Lie groups||--||unique 2-divisibility allows us to halve the Lie bracket or commutator map||uniquely 2-divisible class two case||--||--|
|LCS-Baer correspondence||LCS-Baer Lie rings||LCS-Baer Lie groups||--||can do unique halving within the derived subring or derived subgroup, not necessarily in the whole group||allows us to consider direct products of abelian groups (respectively, abelian Lie rings) and Baer Lie groups (respectively, Baer Lie rings). In particular, this slight generalization covers all finite nilpotent groups whose 2-Sylow subgroup is abelian and all other Sylow subgroups have class at most two.||cyclic group:Z2||2|
|CS-Baer correspondence||CS-Baer Lie rings||CS-Baer Lie groups||--||can do unique halving of elements of derived subring/subgroup within some intermediate subring/subgroup that is still central||(anything more than LCS-Baer?)||Nothing finite||--|
|linear halving generalization of Baer correspondence||Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two||Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two||Lie ring of nilpotency class two||can find a linear half, not necessarily a unique or natural choice||covers some finite and infinite non-abelian 2-groups of nilpotency class two. The smallest example is of order , namely SmallGroup(64,57) on the group side.||SmallGroup(64,57)||64|
|cocycle halving generalization of Baer correspondence||Lie ring arising as the double of a class two Lie cring||group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle||class two Lie cring||can find a skew-symmetric cyclicity-preserving 2-cocycle that functions as the half, not necessarily unique or natural choice.||covers some finite and infinite non-abelian 2-groups of nilpotency class two.||M16 and central product of D8 and Z4||16|
We include here some examples of finite groups of prime power order that do not fall under the Baer correspondence or the LCS-Baer correspondence but fall under this more general correspondence. This means that we only consider finite non-abelian 2-groups. Note that since any finite nilpotent group is a direct product of Sylow subgroups and the correspondence works separately on each Sylow factor, there is no loss of generality in restricting to 2-groups.
Role in explaining 1-isomorphisms
This correspondence plays an important role in explaining 1-isomorphisms between non-abelian groups of nilpotency class two and abelian groups. We list here some cases:
|Order||Number of non-abelian groups||Number of class two non-abelian groups||Total number of non-abelian groups of this order that are 1-isomorphic to abelian groups||Total number of non-abelian groups of this order and class two that are 1-isomorphic to abelian groups||Number of these for which the 1-isomorphism arises from a cocycle skew reversal generalization of Baer correspondence||Number of these for which the 1-isomorphism does not arise from a cocycle skew reversal generalization of Baer correspondence||List of these|
|64||256||117||29||28||18 (or more?)||11 (or less?)|