Baer correspondence

This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
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Definition

The Baer correspondence is a special case of the Lazard correspondence, and is a correspondence as follows:

Uniquely 2-divisible groups of nilpotency class at most two (called Baer Lie groups) $\leftrightarrow$ Uniquely 2-divisible Lie rings of nilpotency class at most 2 (called Baer Lie rings)

For any fixed odd prime number $p$, any p-group is uniquely 2-divisible, and so is any p-Lie ring, so the Baer correspondence restricts to a correspondence:

Class two $p$-groups $\leftrightarrow$ Class two $p$-Lie rings

Each group is 1-isomorphic to the additive group of its corresponding Lie ring, i.e., there is a bijection between them that restricts to an isomorphism on cyclic subgroups.

More explicitly, the Lie ring of any given uniquely 2-divisible class two group can be viewed as the same set with Lie ring operations defined using a fixed formula in terms of group operations, and the Lie group of any given uniquely 2-disible Lie ring is defined using a fixed formula in terms of the Lie ring operations. The procedures for going from a group to a Lie ring and a Lie ring to a group are inverses of each other, and the bijection between a group and its associated Lie ring has a number of nice properties discussed here.

From group to Lie ring

For proof that this construction works, refer: Proof of Baer construction of Lie ring for Baer Lie group

Suppose $G$ is a uniquely 2-divisible class two group. Let $[,]$ denote the commutator of two elements. Note that we can adopt either the left or the right convention -- the two definitions are equal because the group has class two. Denote by ${\sqrt {}}$ the function that takes an element and returns the unique element whose square is that element. If $g$ has finite order $m$, then ${\sqrt {g}}=g^{(m+1)/2}$. We give $G$ the structure of a Lie ring as follows:

Lie ring operation that we need to define	Definition in terms of the group operations	Further comments
Addition, i.e., define $x+y$ for $x,y\in G$	$x+y:={\frac {xy}{\sqrt {[x,y]}}}$	Since $G$ has class two, $[x,y]$ is central. Since center of uniquely p-divisible group is uniquely p-divisible, applied to the prime 2, we get that ${\sqrt {[x,y]}}$ is central. Thus, it makes sense to divide by this element without specifying whether the division occurs on the left or on the right.
Identity element for addition, denoted $0$.	Same as identity element for group multiplication, denoted $e$ or $1$.	This automatically follows from the way addition is defined.
Additive inverse, i.e., define $-x$ for $x\in G$.	Same as $x^{-1}$, i.e., the multiplicative inverse in the group.	This automatically follows from the way addition is defined.
Lie bracket, i.e., the $[,]$ map in the Lie ring.	Same as the group commutator $[x,y]=xyx^{-1}y^{-1}$.

The claim is that with these operations, $G$ acquires the structure of a class two Lie ring.

From Lie ring to group

For proof that this construction works, refer: Proof of Baer construction of Lie group for Baer Lie ring

Suppose $L$ is a uniquely 2-divisible class two Lie ring, with addition denoted $+$ and Lie bracket denoted $[,]$. We give $L$ the structure of a class two group as follows:

Group operation that we need to define	Definition in terms of the Lie ring operations	Further comments
Group multiplication	$xy:=x+y+{\frac {1}{2}}[x,y]$	Since center of uniquely 2-divisible Lie ring is uniquely 2-divisible, we obtain that the element ${\frac {1}{2}}[x,y]$ is central.
Identity element for multiplication	Same as the zero element $0$ of the Lie ring.
Multiplicative inverse $x^{-1}$.	Same as the additive inverse $-x$.
Group commutator $[x,y]=xyx^{-1}y^{-1}$	Same as the Lie bracket $[x,y]$.

Examples

In the case of an abelian group, the corresponding Lie ring is an abelian Lie ring and the additive group of the Lie ring coincides with the original abelian group. In other words, abelian groups correspond to abelian Lie rings.

Groups of prime-cube order

The behavior is the same for all odd primes $p$ for groups of order $p^{3}$.

Group	GAP ID (second part)	Lie ring	Additive group of Lie ring	Description of Baer correspondence
prime-cube order group:U(3,p)	3	upper-triangular nilpotent Lie ring:u(3,p)	elementary abelian group of prime-cube order	Baer correspondence between U(3,p) and u(3,p)
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	(insert name)	direct product of cyclic group of prime-square order and cyclic group of prime order	(insert link)

Groups of prime-fourth order

We first consider groups of order $3^{4}=81$.

Group	GAP ID (second part)	Lie ring	Additive group of Lie ring	Description of Baer correspondence
SmallGroup(81,3)	3	(insert link)	direct product of Z9 and E9	(insert link)
semidirect product of Z9 and Z9	4	(insert link)	direct product of Z9 and Z9	(insert link)
semidirect product of Z27 and Z3	6	(insert link)	direct product of Z27 and Z3	(insert link)
direct product of prime-cube order group:U(3,3) and Z3	12	(insert link)	elementary abelian group:E81	(insert link)
direct product of semidirect product of Z9 and Z3 and Z3	13	(insert link)	direct product of Z9 and E9	(insert link)
central product of prime-cube order group:U(3,3) and Z9	14	(insert link)	direct product of Z9 and E9	(insert link)

Generalizations

There are three kinds of generalizations:

• 2-local Baer correspondence: Generalization to structures other than groups and Lie rings, also relaxing nilpotency class two to 2-local nilpotency class two.
• Generalized Baer correspondence: Generalization to situations where both the group and the Lie ring has nilpotency class two but neither of them is uniquely 2-divisible.
• Lazard correspondence: Generalization to situations of higher nilpotency class, but with the requirement of unique $p$-divisibility for all primes $p$ up to and including the nilpotency class.