ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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A permutation-induced summand is detected as relatively projective by Higman's criterion

Example

Let $H \leq G$ be finite groups and consider the permutation module

$k [G / H] ≅ {Ind}_{H}^{G} (k),$

where $k$ is the trivial $k H$ -module. Then Higman's criterion detects $k [G / H]$ as relatively $H$ -projective.

Facts & Assumptions

Given: A subgroup $H \leq G$ of a finite group and the permutation module $M = k [G / H]$ .

[L1]

Higman's criterion says that a module is relatively $H$ -projective exactly when the identity is a relative trace from an $H$ -endomorphism (Higman's criterion characterizes relative projectivity through the relative trace idempotent test).

Verification

technique · direct

1.1given

The module $M$ is itself induced from the trivial $k H$ -module, so it is relatively $H$ -projective by definition.

2.1L1step 1.1∎

Applying [L1] to the relatively $H$ -projective module of step 1.1 produces an $H$ -endomorphism $α$ with ${Tr}_{H}^{G} (α) = {id}_{M}$ . Thus Higman's criterion detects this permutation-induced module exactly as expected.

Depends on

Higman's criterion characterizes relative projectivity through the relative trace idempotent test

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

J. MacQuarrie, Modular Representations of Profinite Groups (standard reference, not scraped)
Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)