Alphabeta MathComposition Series, the Jordan–Hölder Theorem and Solvable Groups
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✓ 20 results · all verified · 11 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Coxeter Groups
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Subnormal and normal series, factors, refinements, and equivalence
Definition
Composition series, composition factors, and composition length
Theorem
Every finite group has a composition series
Lemma
Dedekind's modular law for subgroup products
Theorem
The Zassenhaus butterfly lemma
Theorem
The Schreier refinement theorem
Theorem
The Jordan–Hölder theorem for groups
Corollary
The order of a finite group is the product of the orders of its composition factors
Definition
Characteristic subgroups
Lemma
Characteristic subgroups are normal, and characteristicity is transitive
Theorem
The derived subgroup is characteristic and the abelianization is universal
Definition
The derived series, solvable groups, and derived length
Lemma
Homomorphisms respect commutator subgroups and derived series
Theorem
A group is solvable if and only if it has a subnormal series with abelian factors
Theorem
Subgroups and quotients of solvable groups are solvable
Theorem
Extensions and finite direct products of solvable groups are solvable
Theorem
A finite group is solvable if and only if all its composition factors are cyclic of prime order
Theorem
A₅ and Sₙ for n≥5 are not solvable
Definition
Subgroup commutators and the lower central series
Definition
The upper central series
Definition
Nilpotent groups and nilpotency class
Lemma
Central factors are equivalent to adjacent commutator containments
Theorem
Nilpotence via central series, the upper central series, and the lower central series
Theorem
Subgroups, quotients, and finite direct products of nilpotent groups are nilpotent
Theorem
Every finite p-group is nilpotent
Corollary
Nilpotent groups, and in particular finite p-groups, are solvable
Theorem