False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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False statement: Maschke's theorem holds over every field

Statement

False claim. If $G$ is a finite group and $V$ is a finite-dimensional representation of $G$ over a field $k$ , then every subrepresentation of $V$ has a $G$ -invariant complement.

Facts & Assumptions

Given: A prime $p$ , the field $k = Z / p$ , the cyclic group $C_{p} = ⟨ g ⟩$ , and the matrix $J = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ .

[L1]

For every prime $p$ , the ring $Z / p$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L2]

A finite-dimensional representation is a group homomorphism into $GL (V)$ , and a subrepresentation is an invariant subspace (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree, Subrepresentations, direct sums of representations, and irreducibility).

Refutation

technique · direct

1.1

Let $N = J - I$ , so $N^{2} = 0$ . A short induction gives $J^{m} = (I + N)^{m} = I + m N$ for every $m \geq 0$ , and in $k = Z / p$ this yields $J^{p} = I$ because $p = 0$ . Therefore sending the generator $g$ to $J$ defines a two-dimensional representation of $C_{p}$ over $k$ in the sense of [L2]. The line $L = k e_{1}$ is invariant because $J e_{1} = e_{1}$ .

L1L2givenalgebra

2.1

Every one-dimensional complement to $L$ has the form $M = k (a e_{1} + e_{2})$ for some $a \in k$ . But $J (a e_{1} + e_{2}) = (a + 1) e_{1} + e_{2} .$ If this vector lay in $M$ , then $(a + 1) e_{1} + e_{2} = λ (a e_{1} + e_{2})$ for some $λ \in k$ . Comparing the $e_{2}$ -coefficients gives $λ = 1$ , and then comparing the $e_{1}$ -coefficients gives $a + 1 = a$ , impossible in a field. So $L$ has no invariant complement. This refutes the claim.

L2step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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, Examples 1.1.4 and 1.1.7 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Example 3.3 (standard reference, not scraped)