Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal \(p\)-Sylow subgroup
Let \(k\) be an algebraically closed field of characteristic \(p > 0\). Let \(G\) be a finite group with a cyclic normal \(p\)-Sylow subgroup \(P\), \(H\) be a complement in \(G\). The Mackey algebra is the basic \(k\)-algebra whose category of finitely generated modules is equivalent to the category of Mackey functors for \(G\) with the property that all \(k\)-spaces involved are finite dimensional over \(k\). The simple Mackey functors for \(G\) are in one-to-one correspondence to the conjugacy classes of pairs \((G',E)\), \(G' \leq G\), \(E\) a simple \(k(N_ G(G')/G')\)-module. The author proves that the simple Mackey functors associated with those subgroups of \(G\) which are not contained in \(Z = P \times C_ H P\) are projective and injective. The simple Mackey functors associated with the subgroups of \(Z\) form in the quiver of the Mackey algebra connected components of a special form. This form is explicitly computed by the author.
scientific article; zbMATH DE number 130635
scientific article