Math 2.0

Group theory and generalizations Definition

Composition series

Builds on Subnormal seriesSimple group

Definition

Let GG be a group. A composition series of GG is a subnormal series {e}=G0⊴G1⊴⋯⊴Gn=G\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G in which each factor Gi/Gi−1G_i / G_{i-1} is a simple group.

Remarks

Equivalently, the inclusions are strict and each Gi−1G_{i-1} is a maximal proper normal subgroup of GiG_i. Each finite group has a composition series, but the group ℤ\mathbb{Z} has none. For example, {0}⊴⟨6⟩⊴⟨3⟩⊴ℤ/12ℤ\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 3 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z} and {0}⊴⟨6⟩⊴⟨2⟩⊴ℤ/12ℤ\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 2 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z} are composition series.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSimple group, definition, no evidence levelSimple groupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionComposition series, definition, no evidence levelComposition seriesdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Conjugate elements definition
  3. Subgroup definition
  4. Conjugate subgroup definition
  5. Left coset definition
  6. Right coset definition
  7. Normal subgroup definition
  8. Simple group definition
  9. Normality via left and right cosets statement
  10. Quotient group definition
  11. Subnormal series definition
  12. Composition series definition

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Composition_series, For groups
About this page
Tag0NB8VPB93V
URIm2://0NB8VPB93V
Kinddefinition
WritingW1 stated
MSC20E15, 20D30
DefinesComposition series
UsesSubnormal series, Simple group

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