Alphabeta MathComposition Series, Solvability and Nilpotence: Examples and Counterexamples
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✓ 10 results · all verified · 6 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
Coxeter Groups
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
Two composition series of C₁₂ have the same factors in different orders
Example
A composition series and the derived series of S₃
Example
Composition and derived series of S₄
Example
The dihedral group of order eight is nilpotent of class two
Example
The integral Heisenberg group is nilpotent of class two
False statement
The composition factors determine a finite group up to isomorphism
False statement
Every solvable group is abelian
False statement
An extension of nilpotent groups is nilpotent
False statement
Every subnormal series is a normal series
False statement