DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-04

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A module is relatively H-projective when it is a direct summand of one induced from H

Definition

Let $G$ be a finite group, let $H \leq G$ , let $k$ be a field, and let $M$ be a $k G$ -module. The module $M$ is relatively $H$ -projective if there exists a $k H$ -module $W$ such that $M$ is a direct summand of ${Ind}_{H}^{G} W$ .

Assuming the Axiom of Choice, when $H = 1$ a basis $B$ of $W$ gives ${Ind}_{1}^{G} W ≅ ⨁_{B} k G$ , which may have infinite rank. Thus a relatively $1$ -projective module is projective. Conversely, the free-summand characterization of Equivalent characterizations of projective modules realizes every projective $k G$ -module as a summand of $(k G)^{(I)} ≅ {Ind}_{1}^{G} (k^{(I)})$ . Hence, under Choice, relative $1$ -projectivity recovers ordinary projectivity in the sense of Projective modules and the lifting property.

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Sources

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