Math 2.0

statement proposition

Subgroups of solvable groups are solvable

Statement

Let GG be a solvable group and let HH be a subgroup of GG. Then HH is solvable.

Remarks

A proof uses the derived series: H(i)⊆G(i)H^{(i)} \subseteq G^{(i)} for each ii, so G(n)={e}G^{(n)} = \{e\} implies H(n)={e}H^{(n)} = \{e\}. More generally, if there is an injective homomorphism from HH into a solvable group, then HH is solvable. For example, each subgroup of S4S_4 is solvable.

Explanations

L1 human

Commutators of elements of HH are also commutators in GG. So the derived series of HH stays inside the derived series of GG. If the larger series reaches the identity, the smaller one does too.

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To read this node, read these 21 nodes in this order. The foundations come first, and the list ends with this node.

Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionConjugate 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 cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionQuotient group, definition, no evidence levelQuotient groupdefinitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionDerived series, definition, no evidence levelDerived seriesdefinitionSolvable group, definition, no evidence levelSolvable groupdefinitionSubgroups of solvable groups are solvable, statement, no evidence levelSubgroups of solvable…statement
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.
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  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Commutator of two elements definition
  5. Conjugate elements definition
  6. Subgroup definition
  7. Abelian group definition
  8. Conjugate subgroup definition
  9. Left coset definition
  10. Right coset definition
  11. Subgroup criterion statement
  12. Normal subgroup definition
  13. Intersection of subgroups is a subgroup statement
  14. Normality via left and right cosets statement
  15. Subgroup generated by a subset definition
  16. Quotient group definition
  17. Commutator subgroup definition
  18. Subnormal series definition
  19. Derived series definition
  20. Solvable group definition
  21. Subgroups of solvable groups are solvable statement

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

Sources

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Properties

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