Alphabeta MathHow statement and proof provenance work
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The derived series, solvable groups, and derived length
Definition
The derived series of a group is defined recursively by Each term is characteristic, hence normal, in the preceding term by The derived subgroup is characteristic and the abelianization is universal.
The group is solvable if for some . Its derived length is the least such . This least index exists because the set of terminating indices is a nonempty subset of and every such subset has a least element (The well-ordering principle). Thus the trivial group has derived length , and a nontrivial abelian group has derived length .
Depends on
Used by
- M-groups are solvable (Taketa) Corollary
- Nilpotent groups, and in particular finite p-groups, are solvable Corollary
- Under the ultrafilter lemma, solvable groups and locally finite groups are amenable Corollary
- A solvable group that is not an M-group Counterexample
- A composition series and the derived series of S₃ Example
- Composition and derived series of S₄ Example
- Every solvable group is abelian False statement
- False statement: finite nilpotent groups and finite solvable groups are the same False statement
- A nontrivial normal subgroup of a solvable group contains a nontrivial abelian subgroup normal in the whole group Lemma
- Homomorphisms respect commutator subgroups and derived series Lemma
- Implications and limits for M-groups Remark
- A group is solvable if and only if it has a subnormal series with abelian factors Theorem
- A₅ and Sₙ for n≥5 are not solvable Theorem
- Burnside's pᵃqᵇ theorem Theorem
- Extensions and finite direct products of solvable groups are solvable Theorem
- Subgroups and quotients of solvable groups are solvable Theorem
Dependency tree · two levels
15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
- K. Conrad, Subgroup Series I (standard reference, not scraped)
- K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)