Cocycle skew reversal generalization of Baer correspondence

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:

• $x * (y + z) + (y * z) = (x + y) * z + (y * z)$ for all $x,y,z \in L$, i.e., $*$ is a 2-cocycle for trivial group action of $L$ on itself.
• $x * y = 0$ if $\langle x,y \rangle$ is cyclic.
• $x * (y * z) = 0$ for all $x,y,z \in L$.
• $[x,y] = (x * y) - (y * x)$ for all $x,y \in L$.

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?)