A characterization of permutation modules. (Q934061)

Let \(k\) be a field of characteristic \(p>0\), let \(P\) be a finite \(p\)-group, and let \(V\) be a finite-dimensional module over the group algebra \(kP\). For a subgroup \(Q\) of \(P\), \(V^Q\) and \(V_Q\) denote the socle and head, respectively, of \(V\). Then \[ V[[Q]]:=V^Q/\Bigl(\sum_{S<Q}\text{Tr}^Q_S(V^Q)+\sum_{Q<T}V^T\Bigr) \] becomes a \(kN_P(Q)\)-module; here \(\text{Tr}^Q_S\colon V^S\to V^Q\) denotes the relative trace map. The author shows that \[ \dim V\leq\sum_Q|P:Q|\dim\bigl(V[[Q]]_{N_P(Q)}\bigr) \] where \(Q\) ranges over a transversal for the conjugacy classes of subgroups of \(P\). Moreover, equality holds if and only if \(V\) is a permutation module. His proof makes use of cohomological Mackey functors.