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 setup of the correspondence

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

Baer Lie groups $\leftrightarrow$ Baer Lie rings

Here:

• A Baer Lie group is a 2-powered (i.e., uniquely 2-divisible) group of nilpotency class at most two
• A Baer Lie ring are 2-powered (i.e., uniquely 2-divisible) Lie ring of nilpotency class at most two

The Baer correspondence preserves underlying sets, i.e., a group and Lie ring that are in Baer correspondence have the same underlying set.

The mapping in the direction from groups to Lie rings will be denoted $\log$ and the mapping in the direction from Lie rings to groups will be denoted $\exp$. Explicitly:

• For any Baer Lie group $G$, we define its Baer Lie ring $\log (G)$ as having the same underlying set and with the Lie ring operations defined using fixed formulas of the group operations.
• For any Baer Lie ring $L$, we define its Baer Lie group $\exp (L)$ as having the same underlying set and with the group operations defined using fixed formulas of the group operations.

The p-group case

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

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 Baer Lie group, i.e., a 2-powered group of nilpotency class (at most) two. 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 the underlying set of $G$ the structure of a Lie ring, denoted $\log (G)$ or $\log G$, 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 $x,y\in G$	$x+y:=\frac{xy}{\sqrt{[x,y]}}=\sqrt{x}y\sqrt{x}=\sqrt{x{y}^{2}x}$	Since $G$ has class two, $[x,y]$ is central. Since center is local powering-invariant, 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. The other two definitions are equivalent, but this requires some algebraic manipulation to show. These definitions are significant because they show that the additive group operation is the same as the twisted multiplication defined for generic 2-powered groups.
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 $[x,y]=xy{x}^{-1}{y}^{-1}$.

The claim is that with these operations, $G$ acquires the structure of a 2-powered class two Lie ring, i.e., a Baer 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 Baer Lie ring, i.e., a uniquely 2-divisible class two Lie ring, with addition denoted $+$ and Lie bracket denoted $[,]$. We give the underlying set of $L$ the structure of a class two group, denoted $\exp L$ or $\exp (L)$, 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 is local powering-invariant in Lie ring, 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 $[x,y]=xy{x}^{-1}{y}^{-1}$	Same as the Lie bracket $[x,y]$.

The claim is that with these operations, $L$ acquires the structure of a 2-powered class two group.

Mutually inverse nature

Further information: Proof of mutual inverse nature of the Baer constructions between group and Lie ring

The two operations described above are two-sided inverses of each other. Explicitly:

1. If we start with a Baer Lie group, construct its Baer Lie ring, and then construct the Baer Lie group of that, we return to the original Baer Lie group. In symbols, $\exp (\log (G))=G$ for any Baer Lie group $G$.
2. If we start with a Baer Lie ring, construct its Baer Lie group, and then construct the Baer Lie ring of that, we return to the original Baer Lie ring. In symbols, $\log (\exp (L))=L$ for any Baer Lie ring $L$.

Functoriality and isomorphism of categories

Given a homomorphism of groups $\phi :G_1\to G_2$ of Baer Lie groups, we can define a homomorphism of Baer Lie rings $\log \phi :\log (G_1)\to \log (G_2)$ between their corresponding Baer Lie rings, such that both homomorphisms are the same as set maps.

Similarly, for a homomorphism $\phi :L_1\to L_2$ of Baer Lie rings, we can define a homomorphism $\exp (\phi ):\exp (L_1)\to \exp (L_2)$ between the corresponding Baer Lie groups.

Thus, $\log$ and $\exp$ can be viewed as functors. Explicitly, the two categories of interest are:

• The category of Baer Lie groups: This is the full subcategory of the category of groups whose objects are the Baer Lie groups. Here, full subcategory means that every morphism in the bigger category between two objects in the subcategory is also in the subcategory. In this case, it means that every group homomorphism between Baer Lie groups is a morphism in the subvategory.
• The category of Baer Lie rings: This is the full subcategory of the category of Lie rings whose objects are the Baer Lie rings.

The functors are as follows:

• $\log$ defines a functor from the category of Baer Lie groups to the category of Baer Lie rings.
• $\exp$ defines a functor from the category of Baer Lie rings to the category of Baer Lie groups.

The functors are two-sided inverses of each other, i.e., $\log \circ \exp$ is the identity functor of the ctegory of Baer Lie rings and $\exp \circ \log$ is the identity functor of the category of Baer Lie groups. Thus, the two categories are isomorphic categories. This isomorphism type of category is termed the Baer Lie category.

Baer correspondence up to isomorphism

A Baer correspondence up to isomorphism between a Baer Lie group $G$ and a Baer Lie ring $L$ can be defined using the following equivalent data:

• An isomorphism of groups from $G$ to $\exp (L)$.
• An isomorphism of Lie rings from $\log (G)$ to $L$.

The Baer correspondence up to isomorphism is often described by specifying the set map from the underlying set of $G$ to the underlying set of $L$, or the set map from the underlying set of $L$ to the underlying set of $G$. Somewhat confusingly, those set maps are referred to as $\log$ and $\exp$ respectively, i.e., $\log :G\to L$ and $\exp :L\to G$ are the set maps. Note that this use of notation differs somewhat from the use of $\log$ and $\exp$ as functors above.

Conceptual interpretation

Analogy with center and radius, or mean and mean deviation

Suppose $a,b\in \mathbb{R}$. The arithmetic mean of $a$ and $b$ is $c=(a+b)/2$ and the mean deviation is $r=|b-a|/2$. Explicitly, $a$ and $b$ are the endpoints of the interval with center $c=(a+b)/2$ and radius $r=|b-a|/2$. The diameter is $|b-a|$.

We can do something similar with geometric means. For $a,b$ positive reals, the geometric mean is $\sqrt{ab}$ and the geometric deviation is $\sqrt{a/b}$ or $\sqrt{b/a}$ (depending on how you measure it).

We can understand the construction of the Lie ring similarly. The idea is that we have a group with a (possibly) noncommutative multiplication. We want to separate out the "commutative" part of the multiplication (which we store as the addition of the Lie ring) from the "noncommutative" part of the multiplication (which we store as the Lie bracket of the Lie ring). Explicitly, noting that $xy$ and $yx$ commute on account of the class being two, we get the following:

• The commutative part of the multiplication can be thought of as obtained by "averaging" out both products. Explicitly, $x+y$ is the geometric mean of $xy$ and $yx$, i.e., it is the unique central value $z$ such that $\frac{xy}{z}=\frac{z}{yx}$. If we work out the formula for this, we obtain that it is $\frac{xy}{\sqrt{[x,y]}}$. Note that we can alternatively describe it as $x+y=\sqrt{x{y}^{2}x}=\sqrt{y{x}^{2}y}$.
• The noncommutative part of the multiplication can be thought of as the quotient of $xy$ and $yx$, which is given as the group commutator $[x,y]$

Twisted product

The Lie ring addition can also be defined as follows:

$x+y:=x^{1/2}y{x}^{1/2}$

This is a special case of the twisted multiplication of a 2-powered group.

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
unitriangular matrix group:UT(3,p)	3	niltriangular matrix Lie ring:NT(3,p)	elementary abelian group of prime-cube order	Baer correspondence between UT(3,p) and NT(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 2-powered.
• 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 (3-local) nilpotency class.
• Baer correspondence up to isoclinism

References

• Groups with Abelian Central Quotient Group by Reinhold Baer, Transactions of the American Mathematical Society, Volume 44,Number 3, Page 357 - 386(November 1938): ^{Gated PDF (JSTOR)More info}