Relative projective module

For a subgroup $H\le G$, 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

For $\alpha \in {\mathrm{End}}_{RH}(M)$, the relative trace is

Let a finite group $G$ act on an algebra $A$ by conjugation and let $H\le G$. The transfer ideal from $H$ isThe notation records both the source subgroup and the ambient fixed-point algebra.

D. Higman's criterion says that an $RG$-module $M$ is relative projective module for $H$ exactly when there is $\alpha \in {\mathrm{End}}_{RH}(M)$ satisfying

If $[G:H]$ is invertible in $R$, every $RG$-module is relatively H-projective. An $RG$-module is then projective exactly when its restriction to $RH$ is projective.

A vertex of an indecomposable $kG$-module is a subgroup minimal among those for which the module is relatively projective. Vertices exist, form one conjugacy class, and are p-groups when $k$ has characteristic $p$. The vertices of the trivial module are the Sylow p-subgroups.