Math 2.0

Group theory and generalizations Definition

Solvable group

Builds on Subnormal seriesAbelian group

Definition

A group GG is solvable when it has 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 abelian.

Remarks

Equivalently, the derived series of GG reaches {e}\{e\} after finitely many steps. A finite group is solvable if and only if its composition factors are all cyclic of prime order. The group S4S_4 is solvable, by the series {e}⊴V⊴A4⊴S4\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4. The name comes from Galois theory: a polynomial equation over ℚ\mathbb{Q} is solvable by radicals if and only if its Galois group is solvable.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionSolvable group, definition, no evidence levelSolvable groupdefinition
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  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Subgroup definition
  6. Abelian group definition
  7. Conjugate subgroup definition
  8. Left coset definition
  9. Right coset definition
  10. Normal subgroup definition
  11. Normality via left and right cosets statement
  12. Quotient group definition
  13. Subnormal series definition
  14. Solvable group definition

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13 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Definition
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WritingW1 stated
MSC20F16
DefinesSolvable group
UsesSubnormal series, Abelian group

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