Group theory and generalizations Definition
Simple group
Builds on Normal subgroup
Definition
A group is simple when and the only normal subgroups of are and .
Remarks
The cyclic groups of prime order are simple, and they are the only abelian simple groups. The alternating group is simple for , and is the smallest simple group that is not abelian. A simple group has only the trivial quotient and itself as quotients, so it cannot be split into smaller pieces by a quotient.
Reading path
To understand this page from the ground up, read these 6 pages in order. The foundations come first.
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- no evidence level
- Group concept
- Conjugate elements definition
- Subgroup definition
- Conjugate subgroup definition
- Normal subgroup definition
- Simple group definition
Where this leads
13 pages build on this one.
- Classification of finite simple groups
- Jordan–Hölder theorem
- Burnside's p^a q^b theorem
- Abelian simple groups have prime order
- Composition series
- Composition factors
- Krull–Schmidt theorem
- Rational equivalence
- Intersection multiplicity
- Fitting's lemma
- Jordan–Hölder theorem for modules
- Algebraic cycle
- Length of a module
Further reading
- Web https://en.wikipedia.org/wiki/Simple_group, lead section
About this page
| Tag | 0C093D1T4M |
|---|---|
| URI | m2://0C093D1T4M |
| Kind | definition |
| Writing | W1 stated |
| MSC | 20E32 |
| Defines | Simple group |
| Uses | Normal subgroup |
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What this page needs for W2
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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