{c}

D. Higman's criterion says that an $RG$-module $M$ is <relative projective module> for $H$ exactly when there is $\alpha\in\operatorname{End}_{RH}(M)$ satisfying
$$
\operatorname{Tr}_H^G(\alpha)=\operatorname{id}_M.
$$

D. Higman criterion

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 diagram
$$
\begin{array}{ccc}
&M&\\
{}_{\widetilde f}\swarrow&&\searrow^{f}\\
X&\xrightarrow{\pi}&Y
\end{array}
$$
with $\pi:X\twoheadrightarrow Y$ an $RG$-map possessing an $RH$-linear section, there is an $RG$-map $\widetilde f:M\to X$ such that $\pi\widetilde f=f$. Equivalently, every $H$-split epimorphism onto $M$ has a $G$-linear section, or
$$
M\mid\operatorname{Ind}_H^G\operatorname{Res}_H^G M.
$$

For $\alpha\in\operatorname{End}_{RH}(M)$ define the <relative trace>
$$
\operatorname{Tr}_H^G(\alpha)
=\sum_{x\in[G/H]}x\alpha x^{-1}.
$$
The <D. Higman criterion> is
$$
\boxed{M\text{ is relatively }H\text{-projective}
\quad\Longleftrightarrow\quad
\operatorname{id}_M\in
\operatorname{Tr}_H^G\operatorname{End}_{RH}(M).}
$$

Solution

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

For a subgroup $H\leq G$, an $RG$-module $M$ is relatively H-projective when every $RG$-epimorphism onto $M$ that splits after restriction to $H$ already splits over $G$. Equivalently, $M$ is a direct summand of
$$
\operatorname{Ind}_H^G\operatorname{Res}_H^G M.
$$


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