Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 2 / b / Solution

Let $S,T$ be module sources at $D$. Since $S\mid M{\downarrow }_D$ and $M\mid T{\uparrow }^G$, Mackey decomposition givesThe module source $S$ has module vertex $D$. By Krull–Schmidt theorem it is a summand of one term, and minimality of its module vertex forces $D\cap {}^{g}D=D$. As both subgroups have order $|D|$, $g\in N_G(D)$. The corresponding term is simply ${}^{g}T$, which is indecomposable. ThusThis proves conjugacy of sources of an indecomposable module at the specified module vertex.