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
- 1 Definition
- 2 Examples
- 3 Metaproperties
- 4 Relation with other properties
- 5 Facts
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 is said to be ambivalent if, for any , there exists such that .
For a finite group , this is equivalent to saying that any representation with character , for all .
- 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 is ambivalent only if .
- 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:
|Cyclic group:Z2||2 (1)|
|Klein four-group||4 (2)|
|Symmetric group:S3||6 (1)|
Here are some relatively less basic/important groups satisfying the property:
|Alternating group:A5||60 (5)|
|Alternating group:A6||360 (118)|
|Dihedral group:D8||8 (3)|
|Quaternion group||8 (4)|
|Symmetric group:S4||24 (12)|
|Symmetric group:S6||720 (763)|
Here are some even more complicated/less basic groups satisfying the property:
|Dihedral group:D16||16 (7)|
|Direct product of D8 and Z2||16 (11)|
|Mathieu group:M9||72 (41)|
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:
|Alternating group:A4||12 (3)|
Here are some even more complicated/less basic groups that do not satisfy the property:
|Semidihedral group:SD16||16 (8)|
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|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.||Rational group|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|