Math 2.0

definition

Composition series

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.

Explanations

L1 human

A composition series cuts a group into layers that cannot be cut further. Each layer is a simple group. It is like the factorisation of an integer into prime numbers.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

What depends on this

2 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Composition_series, For groups

What links here

Used by