DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

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A vertex is a minimal p-subgroup for relative projectivity, and a source is an indecomposable inducing summand there

Definition

Let $G$ be a finite group, let $k$ be a field of characteristic $p$ , and let $M$ be an indecomposable finite-dimensional $k G$ -module.

A vertex of $M$ is a $p$ -subgroup $Q \leq G$ that is minimal, under inclusion, among the subgroups for which $M$ is relatively $Q$ -projective.
A source of $M$ relative to a vertex $Q$ is an indecomposable direct summand $S$ of ${Res}_{Q}^{G} M$ such that $M$ is a direct summand of ${Ind}_{Q}^{G} S$ .

The relative-projectivity notion is the one from A module is relatively H-projective when it is a direct summand of one induced from H, and the Higman trace criterion from Higman's criterion characterizes relative projectivity through the relative trace idempotent test is the main detection tool.

Depends on

Used by

Dependency tree · two levels

6 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)