Ambivalent group: Difference between revisions

Revision as of 02:49, 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
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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 $g \in G$ .

Relation with other properties

Stronger properties

Weaker properties

Group in which every square is a commutator
Group having a class-inverting automorphism: For an ambivalent group, the identity map is itself a class-inverting automorphism.
Group in which every element is automorphic to its inverse
Square-in-derived group

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.
[[satisfies metaproperty::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	ambivalence is conjugacy-closed subgroup-closed	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.

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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 | [[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 finite 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 is ambivalent.
+| [[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.
 |}