Math 2.0

Group theory and generalizations Proposition

Subgroups of solvable groups are solvable

Builds on Solvable groupDerived seriesSubgroup

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.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

Show the path as a graph
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 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. 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

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Properties
About this page
Tag05Y0F9M0TS
URIm2://05Y0F9M0TS
Kindstatement, proposition
WritingW0 stub
MSC20F16
StatusThis copy of the site has no build data. Run m2 status 05Y0F9M0TS.
UsesSolvable group, Derived series, Subgroup

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

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

What links here

Used by