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.
If is finite then
Statement
Let be a field and let be a finite group. Then the group algebra is finite-dimensional over , with
Facts & Assumptions
Given: A field and a finite group .
The module is free on the set , with basis (The group ring of finitely supported formal -linear combinations of group elements).
The dimension of a finite-dimensional vector space is the size of any finite basis (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis).
Proof
By [L1], the set is a basis of indexed by the finite set , so it has exactly elements.
Applying [L2] to that finite basis gives .
Depends on
Used by
- Every irreducible representation of a finite group has degree at most |G| Corollary
- If char k ∤ |G|, then k[G] is a semisimple ring Corollary
- If k is algebraically closed and char k ∤ |G|, then ∑ᵢ (dimₖ Vᵢ)²=|G| Corollary
- False statement: every finite-dimensional algebra over a field is semisimple False statement
- For a finite p-group in characteristic p, the augmentation ideal of the group algebra is nilpotent Lemma
- For a finite group and a field of characteristic p, the group algebra is a symmetric Frobenius algebra via the coefficient of the identity Proposition
- Group algebra and finite-field specializations of the generic Hecke algebra Proposition
Dependency tree · two levels
18 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
- Peter Webb, A Course in Finite Group Representation Theory, Chapter 1 Section 1.1 (standard reference, not scraped)