Ambivalent group: Difference between revisions

Revision as of 17:04, 8 September 2008

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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.

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 $g \in G$ .

Relation with other properties

Stronger properties

Rational group

Weaker properties

Metaproperties

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

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.

That is, if $ϕ : G \to H$ is a quotient map and $x \in H$ , pick any inverse image $x^{'}$ of $x$ in $G$ . Then, there is a $y \in G$ such that $y x^{'} y^{- 1} = x'^{- 1}$ . Then $ϕ (y) x ϕ (y)^{- 1} = x^{- 1}$ .

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group 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.

Finite unions

This group property is finite union-closed: a group which is a finite union of subgroups, each having the property, also has the property[{Category:Finite join-closed group 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.

Facts

Abelianization

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.

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-{{finite group property}}
+{{group property}}
 {{termrelatedto|linear representation theory}}
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 ===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.
-* Every element 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.
-* 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 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 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==