Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 4 / a / Solution

An $RG$-module $M$ is relative projective module for $H$ when it has the lifting property for every $H$-split epimorphism: whenever the solid arrows form a commutative diagramwith $\pi :X\twoheadrightarrow Y$ an $RG$-map possessing an $RH$-linear section, there is an $RG$-map $\tilde{f}:M\to X$ such that $\pi \tilde{f}=f$. Equivalently, every $H$-split epimorphism onto $M$ has a $G$-linear section, or

For $\alpha \in {\mathrm{End}}_{RH}(M)$ define the relative traceThe D. Higman criterion is