{c}

Let $D$ be a p-subgroup of $G$, put $N=N_G(D)$, and let $k$ have characteristic $p$. The <Brauer morphism> intertwines the two <relative traces>:
$$
\operatorname{Br}_D\!\left(\operatorname{Tr}_D^G(a)\right)
=\operatorname{Tr}_D^N\!\left(\operatorname{Br}_D(a)\right)
\qquad(a\in(kG)^D).
$$
Indeed, $D$ acts on $G/D$ by left multiplication. A coset $gD$ is fixed exactly when $g\in N$, and every other orbit has size divisible by $p$. After applying $\operatorname{Br}_D$, the summands belonging to one such orbit are equal, so every nonfixed orbit contributes zero in characteristic $p$; the fixed cosets give the trace from $D$ to $N$.

Brauer morphism and relative trace

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

{title2=$\operatorname{Tr}_H^G$}

For $\alpha\in\operatorname{End}_{RH}(M)$, the relative trace is
$$
\operatorname{Tr}_H^G(\alpha)
=\sum_{g\in[G/H]}g\alpha g^{-1}\in\operatorname{End}_{RG}(M).
$$


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 Incoming links: Relative trace