Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28
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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.

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The regular representation is faithful

Statement

Let G be a finite group and let k be a field. Then the regular representation of G over k is faithful.

Facts & Assumptions

Given: A finite group G and its regular representation on k[G].

[L1]

In the regular representation, g⋅[h]=[gh] for all g,h∈G (The trivial representation, the regular representation, and permutation representations from finite G-sets).

[L2]

A representation is faithful when the only group element acting as the identity linear map is the identity element of the group (Intertwiners, the spaces Hom⁡G(V,W) and End⁡G(V), equivalent representations, and faithful representations).

Proof

technique · direct
1.1

Suppose g∈G acts as the identity in the regular representation. Applying that operator to the basis vector [e] gives [g]=g⋅[e]=[e] by [L1].

L1L2given
2.1

The basis vectors of k[G] are indexed by the elements of G, so [g]=[e] forces g=e. By [L2], this is exactly faithfulness.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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