Difference between revisions of "Proof of Baer construction of Lie ring for Baer Lie group"

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This statement is part of the [[Baer correspondence]], which in turn is a special case of the [[Lazard correspondence]].
 
This statement is part of the [[Baer correspondence]], which in turn is a special case of the [[Lazard correspondence]].
  
Suppose <math>P</math> is a [[fact about::group of nilpotency class two|class two]] <math>p</math>-group, for <math>p</math> an odd prime. Let <math>[ , ]</math> 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 <math>\sqrt{}</math> the function that takes an element and returns the unique element whose square is that element. In particular, <math>\sqrt{g}= g^{(p^k + 1)/2}</math> where <math>p^k</math> is the [[exponent of a group|exponent]] of <math>P</math>. Then, we give <math>P</math> the structure of a Lie ring as follows:
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Suppose <math>G</math> is a [[fact about::Baer Lie group]], i.e., a [[fact about::uniquely 2-divisible|uniquely 2-divisible]] [[fact about::group of nilpotency class two|class two]] group. Let <math>[ , ]</math> 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 <math>\sqrt{}</math> the function that takes an element and returns the unique element whose square is that element. In particular, if an element <math>g</math> has finite order <math>m</math>, then <math>\sqrt{g} = g^{(m+1)/2}</math> (<math>m</math> must be odd).  
  
* The addition is given by:
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<math>G</math> gets the structure of a Lie ring as follows:
  
<math>x + y := \frac{xy}{\sqrt{[x,y]}}</math>
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{| class="sortable" border="1"
* The additive identity for the Lie ring is the group's identity element.
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! Lie ring operation that we need to define !! Definition in terms of the group operations !! Further comments
* Additive inverses are the same as multiplicative inverses in the group.
+
|-
* The Lie bracket is the same as the commutator in the group.
+
| Addition, i.e., define <math>x + y</math> for <math>x,y \in G</math> || <math>x + y := \frac{xy}{\sqrt{[x,y]}} = \sqrt{x}y\sqrt{x} = \sqrt{xy^2x}</math> || Since <math>G</math> has class two, <math>[x,y]</math> is central. Since [[center is local powering-invariant]], applied to the prime 2, we get that <math>\sqrt{[x,y]}</math> is central. Thus, it makes sense to ''divide'' by this element without specifying whether the division occurs on the left or on the right.<br>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 of a 2-powered group|twisted multiplication defined for generic 2-powered groups]].
 +
|-
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| Identity element for addition, denoted <math>0</math>. || Same as identity element for group multiplication, denoted <math>e</math> or <math>1</math>. || This automatically follows from the way addition is defined.
 +
|-
 +
| Additive inverse, i.e., define <math>-x</math> for <math>x \in G</math>. || Same as <math>x^{-1}</math>, i.e., the multiplicative inverse in the group. || This automatically follows from the way addition is defined.
 +
|-
 +
| Lie bracket, i.e., the <math>[ , ]</math> map in the Lie ring. || Same as the ''group'' commutator <math>[x,y] = xyx^{-1}y^{-1}</math>. ||
 +
|}
  
The claim is that with these operations, <math>P</math> acquires the structure of a class two Lie ring.
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Further, this Lie ring is a [[fact about::Baer Lie ring]]: it is uniquely 2-divisible and has class at most two.
  
 
==Related facts==
 
==Related facts==
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* [[Baer correspondence]] -- the full statement.
 
* [[Baer correspondence]] -- the full statement.
* [[Proof of Baer construction of Lie group for odd-order class two p-Lie ring]]
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* [[Proof of Baer construction of Lie group for Baer Lie ring]]
 
* [[Proof of mutual inverse nature of the Baer constructions between group and Lie ring]]
 
* [[Proof of mutual inverse nature of the Baer constructions between group and Lie ring]]
  
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# [[uses::Class two implies commutator map is endomorphism]]
 
# [[uses::Class two implies commutator map is endomorphism]]
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==Proof==
 
==Proof==
  
===Comment on the operation being well-defined===
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Please see the comment, along with the definition of addition, about division by the square root being well defined.
 
+
===Addition is associative===
Note that since <math>P</math> has class two, <math>[x,y]</math> is central, and since the squareroot of any element is a power of it, <math>\sqrt{[x,y]}</math> is central. Hence, ''dividing'' by this squareroot makes sense without specifying whether it is on the left or on the right.
 
  
===Addition is associative===
+
'''Key proof ingredient''': Fact (1).
  
 
'''To prove''': <math>\! (x + y) + z = x + (y + z)</math>
 
'''To prove''': <math>\! (x + y) + z = x + (y + z)</math>
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<math>(x + y) + z = \frac{xyz}{\sqrt{[x,y]}\sqrt{[xy,z]}}</math>
 
<math>(x + y) + z = \frac{xyz}{\sqrt{[x,y]}\sqrt{[xy,z]}}</math>
  
Since the <math>\sqrt{}</math> operation is just a raising to a power, it is a homomorphism on the center, and we can thus rewrite this as:
+
The <math>\sqrt{}</math> operation is a homomorphism on the center, and we can thus rewrite this as:
  
 
<math>\! (x + y) + z = \frac{xyz}{\sqrt{[x,y][xy,z]}} \qquad (\dagger)</math>
 
<math>\! (x + y) + z = \frac{xyz}{\sqrt{[x,y][xy,z]}} \qquad (\dagger)</math>
  
Similarly, the right side of the original associativity expression becomes:
+
Similarly, the right side of theassociativity expression we need to prove becomes:
  
 
<math>\! x + (y + z) = \frac{xyz}{\sqrt{[x,yz][y,z]}} \qquad (\dagger\dagger)</math>
 
<math>\! x + (y + z) = \frac{xyz}{\sqrt{[x,yz][y,z]}} \qquad (\dagger\dagger)</math>
  
Thus, to prove that the original associativity expression holds, it suffces to show that the right sides of <math>(\dagger)</math> and <math>(\dagger\dagger)</math> are equal, which in turn reduces to proving that:
+
Thus, to prove associativity, it suffices to show that the right sides of <math>(\dagger)</math> and <math>(\dagger\dagger)</math> are equal, which in turn reduces to proving that:
  
 
<math>\! [x,y][xy,z] = [x,yz][y,z]</math>
 
<math>\! [x,y][xy,z] = [x,yz][y,z]</math>
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===Addition is commutative===
 
===Addition is commutative===
 +
 +
'''Key proof ingredient''': The fact that <math>[x,y]^{-1} = [y,x]</math>.
  
 
'''To prove''': <math>\! x + y = y + x</math>
 
'''To prove''': <math>\! x + y = y + x</math>
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The left side is <math>[x,y]</math>. The right side is also <math>[x,y]</math>, which follows from the meaning of squareroot and the fact that <math>[y,x] = [x,y]^{-1}</math>.
 
The left side is <math>[x,y]</math>. The right side is also <math>[x,y]</math>, which follows from the meaning of squareroot and the fact that <math>[y,x] = [x,y]^{-1}</math>.
  
===Inverses and identity agree===
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===Agreement of identity and inverses===
 +
 
 +
'''Key proof ingredient''': The commutator vanishes when one of the elements is the identity or when the two elements are inverses of each other.
  
 
'''To prove''': The identity element of the group is an identity element for <math>+</math>, and inverses in the group are inverses for the <math>+</math> operation.
 
'''To prove''': The identity element of the group is an identity element for <math>+</math>, and inverses in the group are inverses for the <math>+</math> operation.
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* Since the identity element <math>e</math> of the group commutes with every element of the group, we have, for all <math>x</math> in the group that <math>x + e = xe = x</math>, and similarly <math>e + x = ex = x</math>. Thus, <math>e</math> is an identity for addition.
 
* Since the identity element <math>e</math> of the group commutes with every element of the group, we have, for all <math>x</math> in the group that <math>x + e = xe = x</math>, and similarly <math>e + x = ex = x</math>. Thus, <math>e</math> is an identity for addition.
 
* Since every element commutes with its multiplicative inverse in the group, we have, for any <math>x</math> in the group, that <math>x + x^{-1} = xx^{-1} = e</math>, which is the identity element for the group and hence also for <math>+</math>. Similarly, <math>x^{-1} + x</math> is the identity element for the group, and hence also for <math>+</math>.
 
* Since every element commutes with its multiplicative inverse in the group, we have, for any <math>x</math> in the group, that <math>x + x^{-1} = xx^{-1} = e</math>, which is the identity element for the group and hence also for <math>+</math>. Similarly, <math>x^{-1} + x</math> is the identity element for the group, and hence also for <math>+</math>.
 
 
===The Lie bracket is additive in the first variable===
 
===The Lie bracket is additive in the first variable===
  
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This follows from the fact that the commutator of any element with itself is the identity element.
 
This follows from the fact that the commutator of any element with itself is the identity element.
 +
 
===The Lie bracket satisfies the Jacobi condition and gives a class two Lie ring===
 
===The Lie bracket satisfies the Jacobi condition and gives a class two Lie ring===
  
 
Since the Lie bracket coincides with the commutator, and the commutator satisfies that <math>[[x,y],z]</math> is trivial for all <math>x,y,z</math>, the Lie bracket also satisfies the same condition. Thus, it satisfies Jacobi's identity and also the condition for class two.
 
Since the Lie bracket coincides with the commutator, and the commutator satisfies that <math>[[x,y],z]</math> is trivial for all <math>x,y,z</math>, the Lie bracket also satisfies the same condition. Thus, it satisfies Jacobi's identity and also the condition for class two.

Latest revision as of 19:18, 30 July 2013

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

Statement

This statement is part of the Baer correspondence, which in turn is a special case of the Lazard correspondence.

Suppose G is a Baer Lie group (?), i.e., 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. In particular, if an element g has finite order m, then \sqrt{g} = g^{(m+1)/2} (m must be odd).

G gets 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]}} = \sqrt{x}y\sqrt{x} = \sqrt{xy^2x} 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] = xyx^{-1}y^{-1}.

Further, this Lie ring is a Baer Lie ring (?): it is uniquely 2-divisible and has class at most two.

Related facts

Other parts of the Baer correspondence

Other generalizations

Facts used

  1. Class two implies commutator map is endomorphism

Proof

Please see the comment, along with the definition of addition, about division by the square root being well defined.

Addition is associative

Key proof ingredient: Fact (1).

To prove: \! (x + y) + z = x + (y + z)

Proof: The left side becomes:

\! (x + y) + z = \frac{\frac{xy}{\sqrt{[x,y]}} \cdot z}{\sqrt{\left[\frac{xy}{\sqrt{[x,y]}},z\right]}}

We know that for c central, [a,bc] = [ac,b] = [a,b]. Since the reciprocal of \sqrt{[x,y]} is central, it can be dropped from inside commutator expressions, and we simplify to:

(x + y) + z = \frac{xyz}{\sqrt{[x,y]}\sqrt{[xy,z]}}

The \sqrt{} operation is a homomorphism on the center, and we can thus rewrite this as:

\! (x + y) + z = \frac{xyz}{\sqrt{[x,y][xy,z]}} \qquad (\dagger)

Similarly, the right side of theassociativity expression we need to prove becomes:

\! x + (y + z) = \frac{xyz}{\sqrt{[x,yz][y,z]}} \qquad (\dagger\dagger)

Thus, to prove associativity, it suffices to show that the right sides of (\dagger) and (\dagger\dagger) are equal, which in turn reduces to proving that:

\! [x,y][xy,z] = [x,yz][y,z]

Now, by fact (1), the left side is [x,y]([x,z][y,z]), and the right side is also ([x,y][x,z])[y,z], and both are equal by associativity in the group, completing the proof.

Addition is commutative

Key proof ingredient: The fact that [x,y]^{-1} = [y,x].

To prove: \! x + y = y + x

Proof: We have:

\! x + y = \frac{xy}{\sqrt{[x,y]}}

and

\! y + x = \frac{yx}{\sqrt{[y,x]}}

It thus suffices to prove that:

\frac{xy}{\sqrt{[x,y]}} = \frac{yx}{\sqrt{[y,x]}}

which is equivalent to proving that:

\frac{xy}{yx} = \frac{\sqrt{[x,y]}}{\sqrt{[y,x]}}

The left side is [x,y]. The right side is also [x,y], which follows from the meaning of squareroot and the fact that [y,x] = [x,y]^{-1}.

Agreement of identity and inverses

Key proof ingredient: The commutator vanishes when one of the elements is the identity or when the two elements are inverses of each other.

To prove: The identity element of the group is an identity element for +, and inverses in the group are inverses for the + operation.

Proof: For this, note that if x,y commute in the group, then x + y = xy. Now:

  • Since the identity element e of the group commutes with every element of the group, we have, for all x in the group that x + e = xe = x, and similarly e + x = ex = x. Thus, e is an identity for addition.
  • Since every element commutes with its multiplicative inverse in the group, we have, for any x in the group, that x + x^{-1} = xx^{-1} = e, which is the identity element for the group and hence also for +. Similarly, x^{-1} + x is the identity element for the group, and hence also for +.

The Lie bracket is additive in the first variable

To prove: [x+y,z] = [x,z] + [y,z]

Proof: We have:

\! [x+y,z] = [\frac{xy}{\sqrt{[x,y]}},z] = [xy,z] = [x,z][y,z]

Also:

\! [x,z] + [y,z] = \frac{[x,z][y,z]}{\sqrt{[[x,z],[y,z]]}} = [x,z][y,z]

Thus, we obtain [x+y,z] = [x,z] + [y,z].

The Lie bracket is additive in the second variable

This is analogous to additivity in the first variable, as shown above.

The Lie bracket is alternating

This follows from the fact that the commutator of any element with itself is the identity element.

The Lie bracket satisfies the Jacobi condition and gives a class two Lie ring

Since the Lie bracket coincides with the commutator, and the commutator satisfies that [[x,y],z] is trivial for all x,y,z, the Lie bracket also satisfies the same condition. Thus, it satisfies Jacobi's identity and also the condition for class two.