Ambivalent group: Difference between revisions

Latest revision as of 03:35, 13 January 2013

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

Definition

Symbol-free definition

A group is said to be ambivalent if every element in it is conjugate to its inverse.

For a finite group,this is equivalent to saying that every character of the group over complex numbers, is real-valued.

An element in a group that is conjugate to its inverse is termed a real element. Thus, a group is ambivalent if and only if all its elements are real elements.

Definition with symbols

A group $G$ is said to be ambivalent if, for any $g \in G$ , there exists $h \in G$ such that $h g h^{- 1} = g^{- 1}$ .

For a finite group $G$ , this is equivalent to saying that any representation $ρ : G \to G L_{n} (C)$ with character $χ$ , $χ (g) \in R$ for all $g \in G$ .

Examples

Extreme examples

The trivial group is ambivalent.

Important families of groups

Symmetric groups are ambivalent: All the symmetric groups are ambivalent.
Classification of ambivalent alternating groups: The alternating group of degree $n$ is ambivalent only if $n \in {1, 2, 5, 6, 10, 14}$ .
Special linear group of degree two is ambivalent iff -1 is a square
Dihedral groups are ambivalent
Generalized dihedral groups are ambivalent

Groups satisfying the property

Here are some basic/important groups satisfying the property:

	GAP ID
Cyclic group:Z2	2 (1)
Symmetric group:S3	6 (1)

Here are some relatively less basic/important groups satisfying the property:

	GAP ID
Alternating group:A6	360 (118)
Dihedral group:D8	8 (3)
Quaternion group	8 (4)
Symmetric group:S4	24 (12)

Here are some even more complicated/less basic groups satisfying the property:

Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

	GAP ID
Alternating group:A4	12 (3)

Here are some even more complicated/less basic groups that do not satisfy the property:

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	ambivalence is not subgroup-closed	It is possible to have a ambivalent group $G$ and a subgroup $H$ of $G$ such that $H$ is not ambivalent.
characteristic subgroup-closed group property	No	ambivalence is not characteristic subgroup-closed	It is possible to have a ambivalent group $G$ and a characteristic subgroup $H$ of $G$ such that $H$ is not ambivalent.
quotient-closed group property	Yes	ambivalence is quotient-closed	If $G$ is an ambivalent group and $H$ is a normal subgroup of $G$ , the quotient group $G / H$ is an ambivalent group.
conjugacy closed subgroup-closed group property	Yes	conjugacy-closed subgroup of ambivalent group is ambivalent	If $G$ is an ambivalent group and $H$ is a conjugacy-closed subgroup of $G$ , then $H$ is ambivalent.
direct product-closed group property	Yes	ambivalence is direct product-closed	If $G_{i}, i \in I$ are all ambivalent groups, so is their external direct product.
union-closed group property	Yes	ambivalence is union-closed	If a group $G$ can be expressed as a union of subgroups $H_{i}, i \in I$ , each of which is ambivalent, then the whole group $G$ is ambivalent.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
rational-representation group
rational group	any two elements generating the same cyclic subgroup are conjugate.	rational implies ambivalent	ambivalent not implies rational	\|FULL LIST, MORE INFO
strongly ambivalent group	every non-identity element is either an involution or a product of two involutions	follows from strongly real implies real	ambivalent not implies strongly ambivalent	\|FULL LIST, MORE INFO
group with two conjugacy classes	there are two conjugacy classes of elements.			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group in which every square is a commutator	every square element is a commutator	ambivalent implies every square is a commutator	every square is a commutator not implies ambivalent	\|FULL LIST, MORE INFO
group having a class-inverting automorphism	there is a class-inverting automorphism: an automorphism that sends every element to the conjugacy class of its inverse element.	For an ambivalent group, the identity automorphism is class-inverting.	class-inverting automorphism not implies ambivalent	\|FULL LIST, MORE INFO
group in which every element is automorphic to its inverse	for any element of the group, there is an automorphism taking that element to its inverse.	(via group having a class-inverting automorphism)	(via group having a class-inverting automorphism)	\|FULL LIST, MORE INFO
square-in-derived group	every square element is in the derived subgroup			\|FULL LIST, MORE INFO

Conjunction with other properties

Property	Meaning	Result of conjunction	Proof
abelian group	any two elements commute; or equivalently, any two conjugate elements are equal.	elementary abelian 2-group	ambivalent and abelian iff elementary abelian 2-group
nilpotent group	admits a central series.	must be a nilpotent ambivalent 2-group	nilpotent and ambivalent implies 2-group
odd-order group	finite group and its order is odd.	trivial group	odd-order and ambivalent implies trivial

Facts

@@ Line 1: / Line 1: @@
-{{finite group property}}
+[[Importance rank::3| ]]
+{{group property}}
 {{termrelatedto|linear representation theory}}
@@ Line 7: / Line 7: @@
 ===Symbol-free definition===
-A [[finite group]] is said to be ambivalent if it satisfies the following equivalent conditions:
+A [[group]] is said to be '''ambivalent''' if every element in it is [[defining ingredient::conjugate elements|conjugat]]e to its inverse.
+For a finite group,this is equivalent to saying that every character of the group over complex numbers, is real-valued.
-* Every element is conjugate to its inverse
+An element in a group that is conjugate to its inverse is termed a [[defining ingredient::real element]]. Thus, a group is ambivalent if and only if all its elements are real elements.
-* Every character of the group over complex numbers, is real-valued
 ===Definition with symbols===
-A [[finite group]] <math>G</math> is said to be '''ambivalent''' if <math>G</math> satisfies the following equivalent conditions:
+A [[group]] <math>G</math> is said to be '''ambivalent''' if, for any <math>g \in G</math>, there exists <math>h \in G</math> such that <math>hgh^{-1} = g^{-1}</math>.
+For a finite group <math>G</math>, this is equivalent to saying that any representation <math>\rho:G \to GL_n(\mathbb{C})</math> with character <math>\chi</math>, <math>\chi(g)\in \mathbb{R}</math> for all <math>g \in G</math>.
-* For any <math>g \in G</math>, there exists <math>h \in G</math> such that <math>hgh^{-1} = g^{-1}</math>.
+==Examples==
-* For any representation <math>\rho:G \to GL_n(\mathbb{C})</math> with character <math>\chi</math>, <math>\chi(g)\in \mathbb{R}</math> for all <math>g \in G</math>.
-==Relation with other properties==
+===Extreme examples===
+* The [[trivial group]] is ambivalent.
+===Important families of groups===
+* [[Symmetric groups are ambivalent]]: All the symmetric groups are ambivalent.
+* [[Classification of ambivalent alternating groups]]: The alternating group of degree <math>n</math> is ambivalent only if <math>n \in \{ 1,2,5,6,10,14 \}</math>.
+* [[Special linear group of degree two is ambivalent iff -1 is a square]]
+* [[Dihedral groups are ambivalent]]
+* [[Generalized dihedral groups are ambivalent]]
+===Groups satisfying the property===
+{{groups satisfying property sorted by importance rank}}
-===Stronger properties===
+===Groups dissatisfying the property===
-* [[Rational group]]
+{{groups dissatisfying property sorted by importance rank}}
 ==Metaproperties==
-{{Q-closed}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[ambivalence is not subgroup-closed]] || It is possible to have a ambivalent group <math>G</math> and a [[subgroup]] <math>H</math> of <math>G</matH> such that <math>H</math> is not ambivalent.
+|-
+| [[dissatisfies metaproperty::characteristic subgroup-closed group property]] || No || [[ambivalence is not characteristic subgroup-closed]] || It is possible to have a ambivalent group <math>G</math> and a [[characteristic subgroup]] <math>H</math> of <math>G</matH> such that <math>H</math> is not ambivalent.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[ambivalence is quotient-closed]] || If <math>G</math> is an ambivalent group and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> is an ambivalent group.
+|-
+| [[satisfies metaproperty::conjugacy closed subgroup-closed group property]] || Yes || [[conjugacy-closed subgroup of ambivalent group is ambivalent]] || If <math>G</math> is an ambivalent group and <math>H</math> is a [[conjugacy-closed subgroup]] of <math>G</math>, then <math>H</math> is ambivalent.
+|-
+| [[satisfies metaproperty::direct product-closed group property]] || Yes || [[ambivalence is direct product-closed]] || If <math>G_i, i \in I</math> are all [[ambivalent group]]s, so is their [[external direct product]].
+|-
+| [[satisfies metaproperty::union-closed group property]] || Yes || [[ambivalence is union-closed]] || If a group <math>G</math> can be expressed as a union of subgroups <math>H_i, i \in I</math>, each of which is ambivalent, then the whole group <math>G</math> is ambivalent.
+|}
-Any quotient of an ambivalent group is ambivalent. This follows from the fact that given any element in the quotient group, we can conjugate it to its inverse by looking at the image of the conjugating element under the quotient map.
+==Relation with other properties==
-That is, if <math>\phi:G \to H</math> is a quotient map and <math>x \in H</math>, pick any inverse image <math>x'</math> of <math>x</math> in <math>G</math>. Then, there is a <math>y \in G</math> such that <math>yx'y^{-1} = x'^{-1}</math>. Then <math>\phi(y)x\phi(y)^{-1} = x^{-1}</math>.
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::rational-representation group]] || || || ||
+|-
+| [[Weaker than::rational group]] || any two elements generating the same cyclic subgroup are conjugate. ||[[rational implies ambivalent]] || [[ambivalent not implies rational]] || {{intermediate notions short|ambivalent group|rational group}}
+|-
+| [[Weaker than::strongly ambivalent group]] || every non-identity element is either an [[involution]] or a product of two [[involution]]s || follows from [[strongly real implies real]] || [[ambivalent not implies strongly ambivalent]] || {{intermediate notions short|ambivalent group|strongly ambivalent group}}
+|-
+| [[Weaker than::group with two conjugacy classes]] || there are two conjugacy classes of elements. || || || {{intermediate notions short|ambivalent group|group with two conjugacy classes}}
+|}
-{{DP-closed}}
+===Weaker properties===
-Any direct product of ambivalent groups is ambivalent. This follows from the fact that both the relation of being conjugate and the inverse map can be checked coordinate-wise for a direct product.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::group in which every square is a commutator]] || every [[square element]] is a [[commutator]] || [[ambivalent implies every square is a commutator]] || [[every square is a commutator not implies ambivalent]] || {{intermediate notions short|group in which every square is a commutator|ambivalent group}}
+|-
+| [[Stronger than::group having a class-inverting automorphism]] || there is a [[class-inverting automorphism]]: an automorphism that sends every element to the conjugacy class of its inverse element. || For an ambivalent group, the identity automorphism is class-inverting. || [[class-inverting automorphism not implies ambivalent]] || {{intermediate notions short|group having a class-inverting automorphism|ambivalent group}}
+|-
+| [[Stronger than::group in which every element is automorphic to its inverse]] || for any element of the group, there is an automorphism taking that element to its inverse. || (via group having a class-inverting automorphism) || (via group having a class-inverting automorphism) || {{intermediate notions short|group in which every element is automorphic to its inverse|ambivalent group}}
+|-
+| [[Stronger than::square-in-derived group]] || every [[square element]] is in the [[derived subgroup]] || || || {{intermediate notions short|square-in-derived group|ambivalent group}}
+|}
-{{finite union-closed}}
+===Conjunction with other properties===
-A group which is a union of finitely many subgroups, each of which is ambivalent, is also ambivalent. Finiteness of the union is only required so that the big group is itself finite; the condition of every element being conjugate to its inverse is preserved upon arbitrary unions.
+{| class="sortable" border="1"
+! Property !! Meaning !! Result of conjunction !! Proof
+|-
+| [[abelian group]] || any two elements commute; or equivalently, any two conjugate elements are equal. || [[elementary abelian 2-group]] || [[ambivalent and abelian iff elementary abelian 2-group]]
+|-
+| [[nilpotent group]] || admits a [[central series]]. || must be a nilpotent ambivalent 2-group || [[nilpotent and ambivalent implies 2-group]]
+|-
+| [[odd-order group]] || [[finite group]] and its [[order of a group|order]] is odd. || [[trivial group]]  || [[odd-order and ambivalent implies trivial]]
+|}
 ==Facts==
-===Abelianization===
+* [[Center of ambivalent group is elementary abelian 2-group]]
+* [[Abelianization of ambivalent group is elementary abelian 2-group]]
-Since every element of an ambivalent group is conjugate to its inverse, the image of any element in the [[Abelianization]] equals the image of its inverse. Hence, the Abelianization must be a group of exponent two, or equivalently, it must be a direct power of the cyclic group of order two.