Cocycle skew reversal generalization of Baer correspondence

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Statement

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:

Lie ring arising as the skew of a class two near-Lie cring \leftrightarrow group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle

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 L is a Lie ring arising as the skew of a class two near-Lie cring, i.e., there exists a binary operation *: L \times L \to L such that:

Then, we can define a group structure on L in terms of * as follows:

Group operation Denoted as ... Definition in terms of Lie ring operations
Multiplication (x,y) \mapsto xy xy := x + y + (x * y)
Identity element e e := 0
Inverse {}^{-1}, i.e., x \mapsto x^{-1} x^{-1} := -x

It turns out that the group commutator xyx^{-1}y^{-1} is the same as the Lie bracket [x,y] with these operations.

From group to Lie ring

Suppose G 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 \circ:G \times G \to G such that:

  • x \circ y \in Z(G) for all x,y \in G.
  • (x \circ (yz))(y \circ z) = ((xy) \circ z)(x \circ y) for all x,y,z \in G.
  • x \circ y is the identity element whenever \langle x,y \rangle is cyclic.
  • x \circ (y \circ z) is the identity element for all x,y,z \in G.
  • xyx^{-1}y^{-1} = (x \circ y)(y \circ x)^{-1} for all x,y \in G.

Then we can define a Lie ring structure on G as follows:

Lie ring operation Denoted as ... Definition in terms of group operations
Addition (x,y) \mapsto x + y x + y := xy(x \circ y)^{-1}
Lie bracket [x,y] [x,y] := xyx^{-1}y^{-1}, i.e., same as the commutator in the group.
Negative x \mapsto -x -x := x^{-1}
Zero 0 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 2^6, 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

Particular cases

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.

Order Non-abelian group GAP ID (2nd part) Additive group of Lie ring GAP ID (2nd part) Most restrictive correspondence form Description of 1-isomorphism Best cohomology perspective 1 Best cohomology perspective 2 Alternative cohomology perspective
16 central product of D8 and Z4 13 direct product of Z4 and V4 10 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between central product of D8 and Z4 and direct product of Z4 and V4 second cohomology group for trivial group action of V4 on Z4#Generalized Baer Lie rings second cohomology group for trivial group action of E8 on Z2#Direct sum decomposition
16 M16 6 direct product of Z8 and Z2 5 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between M16 and direct product of Z8 and Z2 second cohomology group for trivial group action of V4 on Z4#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2#Generalized Baer Lie rings
32 M32 17 direct product of Z16 and Z2 16 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between M32 and direct product of Z16 and Z2 second cohomology group for trivial group action of V4 on Z8#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2
32 semidirect product of Z8 and Z4 of M-type 4 direct product of Z8 and Z4 3 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between semidirect product of Z8 and Z4 of M-type and direct product of Z8 and Z4 second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2
32 direct product of M16 and Z2 37 direct product of Z8 and V4 36 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between direct product of M16 and Z2 and direct product of Z8 and V4 second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2
32 central product of D8 and Z8 38 direct product of Z8 and V4 36 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between central product of D8 and Z8 and direct product of Z8 and V4 second cohomology group for trivial group action of V4 on Z8#Generalized Baer Lie rings second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and V4 on Z2
32 SmallGroup(32,24) 24 direct product of Z4 and Z4 and Z2 21 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between SmallGroup(32,24) and direct product of Z4 and Z4 and Z2 second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings  ?
32 direct product of SmallGroup(16,13) and Z2 48 direct product of E8 and Z4 45 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring generalized Baer correspondence between direct product of SmallGroup(16,13) and Z2 and direct product of E8 on Z4 second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings  ?
32 SmallGroup(32,2) 2 direct product of Z4 and Z4 and Z2 21 cocycle skew reversal generalization of Baer correspondence, the intermediate object being a class two near-Lie cring generalized Baer correspondence between SmallGroup(32,2) and direct product of Z4 and Z4 and Z2 second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2#Generalized Baer Lie rings -- --
64 semidirect product of Z8 and Z8 of M-type 3 direct product of Z8 and Z8 2 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 semidirect product of Z16 and Z4 of M-type 27 direct product of Z16 and Z4 26 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 M64 51 direct product of Z32 and Z2 50 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 SmallGroup(64,57) 57 direct product of Z4 and Z4 and Z4 55 linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring
64 direct product of SmallGroup(32,4) and Z2 84 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 direct product of M16 and Z4 85 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 central product of M16 and Z8 over common Z2 86 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 112 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 direct product of M32 and Z2 184 direct product of Z16 and V4 183 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 central product of D8 and Z16 185 direct product of Z16 and V4 183 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 direct product of SmallGroup(32,24) and Z2 195 direct product of Z4 and Z4 and V4 192 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 direct product of SmallGroup(16,13) and Z4 198 direct product of Z4 and Z4 and V4 192 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 direct product of M16 and V4 247 direct product of Z8 and E8 246 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 SmallGroup(64,248) 248 direct product of Z8 and E8 246 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 249 direct product of Z8 and E8 246 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 direct product of SmallGroup(16,13) and V4 263 direct product of E16 and Z4 260 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
64 266 direct product of E16 and Z4 260 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring

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
2 0 0 0 0 0 0 --
4 0 0 0 0 0 0 --
8 2 2 0 0 0 0 --
16 9 6 2 2 2 0 --
32 44 26 8 8 7 1 SmallGroup(32,33)
64 256 117 29 28 18 (or more?) 11 (or less?)