Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Supersolvable groups and monomial characters
Definition
A finite group is supersolvable if it has a normal series whose factors have prime order. Here “normal series” means that every is normal in , not merely in (Subnormal and normal series, factors, refinements, and equivalence). An irreducible complex character of is monomial if for some subgroup and linear character of ; is monomial if all its irreducible complex characters are monomial.
Depends on
- Subnormal and normal series, factors, refinements, and equivalence
- Normal subgroup: invariance under conjugation
- The quotient group $G/N$ and coset product $(gN)(hN)=ghN$
- Subrepresentations, direct sums of representations, and irreducibility
- The induced character $\operatorname{Ind}_H^G\chi$ of a complex character
Used by
- A nonabelian supersolvable group has a noncentral abelian normal subgroup, also in its nonabelian quotients Lemma
- A nonabelian supersolvable group has a noncentral normal abelian subgroup Lemma
- Elementary groups are supersolvable Lemma
- Monomiality lifts along a quotient Lemma
- A faithful irreducible is induced from a proper inertia subgroup Proposition
- Finite supersolvable groups are M-groups Theorem
- Finite supersolvable groups are monomial Theorem
- Virtual characters are integrally generated by monomial characters from elementary subgroups Theorem
Dependency tree · two levels
13 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
- Tammo tom Dieck, Representation Theory, Section 4.3 (standard reference, not scraped)