CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

How 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 $char k ∤ ∣ G ∣$ , every finite-dimensional representation of $G$ is completely reducible

Statement

Let $G$ be a finite group, let $k$ be a field with $char k ∤ ∣ G ∣$ , and let $V$ be a finite-dimensional representation of $G$ over $k$ . Then $V$ is completely reducible.

Facts & Assumptions

Given: A finite group $G$ , a field $k$ with $char k ∤ ∣ G ∣$ , and a finite-dimensional representation $V$ of $G$ over $k$ .

[L1]

A representation is completely reducible exactly when its underlying space is an internal direct sum of irreducible subrepresentations, and the zero representation is included by the empty direct sum (A completely reducible representation as a finite direct sum of irreducible subrepresentations).

[L2]

A representation is irreducible when it is nonzero and has no proper nonzero subrepresentation (Subrepresentations, direct sums of representations, and irreducibility).

[L3]

Under the characteristic hypothesis, every subrepresentation has a $G$ -invariant complement (Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$ ).

[A1]

Every nonempty set of positive integers has a least element.

Proof

technique · induction

1.1

If $V = 0$ , then [L1] makes $V$ completely reducible as the empty direct sum.

L1base

1.2

Assume now that $\dim_{k} V > 0$ , and as induction hypothesis suppose every representation of smaller dimension is completely reducible. Among the nonzero subrepresentations of $V$ , choose one with least positive dimension, using [A1]. It is irreducible by [L2], since any proper nonzero subrepresentation would have smaller positive dimension. Call this irreducible subrepresentation $W$ . By [L3], there is a subrepresentation $U$ with $V = W \oplus U$ .

A1L2L3givenihchoose

2.1

The subrepresentation $W$ is nonzero, so $\dim_{k} U < \dim_{k} V$ . The induction hypothesis therefore makes $U$ completely reducible. Using [L1] to expand that decomposition and adjoining the irreducible summand $W$ , one gets a direct-sum decomposition of $V$ into irreducible subrepresentations. Hence $V$ is completely reducible.

step 1.2L1discharge-induction∎

Depends on

Used by

Dependency tree · two levels

7 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, Corollary 1.2.5 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Theorem 3.1(i) (standard reference, not scraped)

If char⁡k∤∣G∣chark∤∣G∣, every finite-dimensional representation of GG is completely reducible