Set
$$
\alpha=[G:H]^{-1}\operatorname{id}_M\in\operatorname{End}_{RH}(M).
$$
Every conjugate of $\alpha$ equals $\alpha$, so
$$
\operatorname{Tr}_H^G(\alpha)
=[G:H]\alpha=\operatorname{id}_M.
$$
The <D. Higman criterion> therefore gives
$$
\boxed{\text{every }RG\text{-module is relatively }H\text{-projective}.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-138/4/b/i/solution