Math 2.0

Group theory and generalizations Definition

Composition factors

Builds on Composition series

Definition

Let GG be a group with a composition series {e}=G0⊴G1⊴⋯⊴Gn=G\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G. The composition factors of GG are the simple groups Gi/Gi−1G_i / G_{i-1} for i=1,…,ni = 1, \dots, n, counted with multiplicity. The number nn is the composition length of GG.

Remarks

The Jordan–Hölder theorem shows that the composition factors do not depend on the series, up to order and isomorphism. The composition factors of ℤ/12ℤ\mathbb{Z}/12\mathbb{Z} are ℤ/2ℤ\mathbb{Z}/2\mathbb{Z}, ℤ/2ℤ\mathbb{Z}/2\mathbb{Z} and ℤ/3ℤ\mathbb{Z}/3\mathbb{Z}. The composition factors of S4S_4 are ℤ/2ℤ\mathbb{Z}/2\mathbb{Z}, ℤ/3ℤ\mathbb{Z}/3\mathbb{Z}, ℤ/2ℤ\mathbb{Z}/2\mathbb{Z} and ℤ/2ℤ\mathbb{Z}/2\mathbb{Z}. Groups that are not isomorphic can have the same composition factors, for example ℤ/4ℤ\mathbb{Z}/4\mathbb{Z} and ℤ/2ℤ×ℤ/2ℤ\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 13 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 seriesdefinitionComposition factors, definition, no evidence levelComposition factorsdefinition
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.
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  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
  13. Composition factors definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Composition_series, For groups
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WritingW0 stub
MSC20E15, 20D30
DefinesComposition factors
UsesComposition series

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