Math 2.0

Group theory and generalizations Proposition

Extensions of solvable groups are solvable

Builds on Solvable groupDerived seriesQuotients of solvable groups are solvableSubgroups of solvable groups are solvable

Statement

Let GG be a group and let NN be a normal subgroup of GG. If NN and G/NG/N are solvable, then GG is solvable.

Remarks

If (G/N)(k)(G/N)^{(k)} is trivial, then G(k)⊆NG^{(k)} \subseteq N, so G(k+m)⊆N(m)G^{(k+m)} \subseteq N^{(m)} for each mm. Choose mm with N(m)={e}N^{(m)} = \{e\}; then G(k+m)={e}G^{(k+m)} = \{e\}. With subgroups and quotients, this gives: GG is solvable if and only if NN and G/NG/N are solvable. In particular, direct products and semidirect products of solvable groups are solvable.

Reading path

To understand this page from the ground up, read these 23 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 groupdefinitionQuotients of solvable groups are solvable, statement, no evidence levelQuotients of solvable…statementSubgroups of solvable groups are solvable, statement, no evidence levelSubgroups of solvable…statementExtensions of solvable groups are solvable, statement, no evidence levelExtensions of solvabl…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. Quotients of solvable groups are solvable statement
  22. Subgroups of solvable groups are solvable statement
  23. Extensions of solvable groups are solvable statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Properties
About this page
Tag01EY0EM7RP
URIm2://01EY0EM7RP
Kindstatement, proposition
WritingW0 stub
MSC20F16
StatusThis copy of the site has no build data. Run m2 status 01EY0EM7RP.
UsesSolvable group, Derived series, Quotients of solvable groups are solvable, Subgroups of solvable groups are solvable

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