Baer correspondence

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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.
View other such facts for p-groups|View other such facts for finite groups

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 Baer Lie rings

Here:

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 and the mapping in the direction from Lie rings to groups will be denoted . Explicitly:

  • For any Baer Lie group , we define its Baer Lie ring 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 , we define its Baer Lie group 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 , any p-group is uniquely 2-divisible, and so is any p-Lie ring, so the Baer correspondence restricts to a correspondence:

Class two -groups Class two -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 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 the function that takes an element and returns the unique element whose square is that element. If has finite order , then . We give the underlying set of the structure of a Lie ring, denoted or , as follows:

Lie ring operation that we need to define Definition in terms of the group operations Further comments
Addition, i.e., define for Since has class two, is central. Since center is local powering-invariant, applied to the prime 2, we get that 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 . Same as identity element for group multiplication, denoted or . This automatically follows from the way addition is defined.
Additive inverse, i.e., define for . Same as , 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 .

The claim is that with these operations, 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 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 the structure of a class two group, denoted or , as follows:

Group operation that we need to define Definition in terms of the Lie ring operations Further comments
Group multiplication Since center is local powering-invariant in Lie ring, we obtain that the element is central.
Identity element for multiplication Same as the zero element of the Lie ring.
Multiplicative inverse . Same as the additive inverse .
Group commutator Same as the Lie bracket .

The claim is that with these operations, 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, for any Baer Lie group .
  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, for any Baer Lie ring .

Functoriality and isomorphism of categories

Given a homomorphism of groups of Baer Lie groups, we can define a homomorphism of Baer Lie rings between their corresponding Baer Lie rings, such that both homomorphisms are the same as set maps.

Similarly, for a homomorphism of Baer Lie rings, we can define a homomorphism between the corresponding Baer Lie groups.

Thus, and 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 subcategory.
  • 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:

  • defines a functor from the category of Baer Lie groups to the category of Baer Lie rings.
  • 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., is the identity functor of the ctegory of Baer Lie rings and 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 and a Baer Lie ring can be defined using the following equivalent data:

  • An isomorphism of groups from to .
  • An isomorphism of Lie rings from to .

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

Conceptual interpretation

Analogy with center and radius, or mean and mean deviation

Suppose . The arithmetic mean of and is and the mean deviation is . Explicitly, and are the endpoints of the interval with center and radius . The diameter is .

We can do something similar with geometric means. For positive reals, the geometric mean is and the geometric deviation is or (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 and 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, is the geometric mean of and , i.e., it is the unique central value such that . If we work out the formula for this, we obtain that it is . Note that we can alternatively describe it as .
  • The noncommutative part of the multiplication can be thought of as the quotient of and , which is given as the group commutator

Twisted product

The Lie ring addition can also be defined as follows:

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 for groups of order .

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 .

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:

References