Group theory and generalizations Foundations Concept
Normal subgroup
Summary
A normal subgroup of a group is a subgroup that is equal to each of its conjugates. The normal subgroups are exactly the subgroups that give quotient groups, and exactly the kernels of homomorphisms.
Where this leads
35 pages build on this one.
- Zassenhaus lemma
- Pro-p group
- Z* theorem
- Profinite completion
- Block system
- Schur–Zassenhaus theorem
- Fitting subgroup
- Supersolvable group
- Loop theorem
- Hall's theorem on solvable groups
- Primitive permutation group
- Quotient of a topological group
- Margulis normal subgroup theorem
- Residually finite group
- Subnormal subgroup
- Hall subgroup
- Sylow theorems
- Restricted Burnside problem
- Galois correspondence for coverings
- Chevalley's structure theorem
- Identity component
- Schreier refinement theorem
- Seifert–van Kampen theorem
- Hopf's formula
- Clifford theory
- Hilbert's eighteenth problem
- HNN extension
- Lyndon–Hochschild–Serre spectral sequence
- Reductive group
- Gleason–Yamabe theorem
- Frobenius's theorem on Frobenius kernels
- Jordan–Schur theorem
- Bieberbach theorems
- Frobenius group
- O'Nan–Scott theorem
Further reading
- Web https://en.wikipedia.org/wiki/Normal_subgroup, Definitions
About this page
| Tag | 0CB0DX5BVV |
|---|---|
| URI | m2://0CB0DX5BVV |
| Aliases | invariant-subgroup |
| Kind | concept |
| Writing | W2 usable |
| MSC | 20A05 |
| Specializes | Subgroup |
| Same as |
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- section Motivation: the question that the notion answers (0 of 1)
- section History: who introduced the notion and when (0 of 1)
- explanations for the audience (0 of 3)
- a person confirmed each agent-written section: rule P04 (0 of 1)
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Results about this
Special cases
Definitions
Used by
- Zassenhaus lemma
- Pro-p group
- Z* theorem
- Profinite completion
- Block system
- Schur–Zassenhaus theorem
- Fitting subgroup
- Supersolvable group
- Loop theorem
- Primitive permutation group
- Quotient of a topological group
- Margulis normal subgroup theorem
- Residually finite group
- Subnormal subgroup
- Sylow theorems
- Restricted Burnside problem
- Galois correspondence for coverings
- Chevalley's structure theorem
- Identity component
- Seifert–van Kampen theorem
- Hopf's formula
- Clifford theory
- HNN extension
- Lyndon–Hochschild–Serre spectral sequence
- Reductive group
- Gleason–Yamabe theorem
- Frobenius's theorem on Frobenius kernels
- Jordan–Schur theorem
- Bieberbach theorems
- O'Nan–Scott theorem
Instances
- The special linear group is the kernel of the determinant
- The even integers form a subgroup of the integers
- The alternating group A3 is normal in S3