Group theory and generalizations Definition
Conjugate elements
Builds on Group
Definition
Let be a group and let . The element is conjugate to when there is with . The element is called the conjugate of by .
Remarks
Conjugacy is an equivalence relation on ; its classes are the conjugacy classes. In an abelian group, , so each element is conjugate only to itself. Conjugate elements have the same order. In the group of invertible matrices, conjugate matrices are similar matrices.
Reading path
To understand this page from the ground up, read these 2 pages in order. The foundations come first.
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- Group concept
- Conjugate elements definition
Where this leads
109 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Semidirect product
- Normality via left and right cosets
- Extensions of solvable groups are solvable
- Quotients of solvable groups are solvable
- Stabilisers of points in the same orbit are conjugate
- Inner automorphism group is isomorphic to G modulo its centre
- Subgroups of solvable groups are solvable
- Classification of finite simple groups
- Finite p-groups are nilpotent
- Class equation
- Subnormal series
- Fixed points of a p-group action
- Second Sylow theorem
- The symmetric group S3
- Outer automorphism group
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- If G modulo its centre is cyclic then G is abelian
- Simplicity of the alternating group
- Normaliser
- Action on left cosets
- Simple group
- The kernel is a normal subgroup
- First isomorphism theorem
- Conjugate subgroup
- Upper central series
- Nilpotent group
- Jordan–Hölder theorem
- Orbit-stabiliser theorem
- Universal property of the abelianisation
- A subgroup whose index is the smallest prime divisor of the order is normal
- Burnside's p^a q^b theorem
- Universal property of the quotient group
- The alternating group A5 is simple
- Nilpotent groups are solvable
- Inner automorphism
- Abelianisation
- Conjugacy classes of the symmetric group correspond to cycle types
- The symmetric group on five or more letters is not solvable
- Abelian simple groups have prime order
- Cauchy's theorem
- Subgroups of abelian groups are normal
- The number of Sylow subgroups divides the index
- A quotient is abelian iff the subgroup contains the commutator subgroup
- Composition series
- Solvable group
- Action by conjugation
- Second isomorphism theorem
- Internal direct product
- Conjugacy class
- Third Sylow theorem
- Von Dyck's theorem
- Frattini argument
- Presentation of a group
- Abelian groups are solvable
- Composition factors
- Torsion subgroup
- Structure theorem for finitely generated abelian groups
- A nontrivial finite p-group has nontrivial centre
- Size of an orbit times the size of the stabiliser
- Correspondence theorem
- Characteristic subgroups are normal
- Order of the alternating group
- Normal core
- Third isomorphism theorem
- Product of a subgroup and a normal subgroup is a subgroup
- Subgroups of index two are normal
- First Sylow theorem
- Size of a conjugacy class is the index of the centraliser
- Normal subgroup
- Groups of order pq
- Integers modulo n as a quotient group
- Free abelian group
- Quotient group
- Burnside's lemma
- Abel–Ruffini theorem
- Zassenhaus lemma
- Galois group of a prime degree polynomial with two nonreal roots
- Proof that subgroups of index two are normal
- The integers modulo the even integers
- The even integers form a subgroup of the integers
- Hall's theorem on solvable groups
- Krull–Schmidt theorem
- Rational equivalence
- Proof that the kernel is a normal subgroup
- Intersection multiplicity
- Hall subgroup
- Fitting's lemma
- Sylow theorems
- Frattini subgroup
- Lattice-ordered group
- Proof of the first isomorphism theorem
- Jordan–Hölder theorem for modules
- The cosets of a subgroup that is not normal do not form a group
- Algebraic cycle
- Algebraic K-theory of schemes
- Structure theorem for compactly generated LCA groups
- Left-orderable group
- Schreier refinement theorem
- Solvability of x^n - a
- Quintic equation
- Shafarevich's theorem on solvable Galois groups
- Length of a module
- Galois' theorem on solvability
- Torsion-free group
- Divisor class group
- Solvable by radicals
- Orderable group
Further reading
About this page
| Tag | 0D1VWQGZZ7 |
|---|---|
| URI | m2://0D1VWQGZZ7 |
| Aliases | conjugacy |
| Kind | definition |
| Writing | W0 stub |
| MSC | 20E45 |
| Defines | Conjugate elements |
| Uses | Group |
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