The modular version of Maschke's theorem for normal abelian \(p\)-Sylows
The authors prove the Donald-Flanigan conjecture in characteristic \(p\) for groups having a normal abelian Sylow \(p\)-subgroup. Let \(p\) be a prime number, let \(G\) be a finite group and let \(k\) be an algebraically closed field of characteristic \(p\). Then, Donald and Flanigan conjectured that there is a deformation of the group algebra \(kG\) to a separable algebra for which the dimensions of the matrix algebras which occur are the same as for the group algebra over the complex numbers. Donald and Flanigan proved this in the abelian case and the second author of the paper under review proved the conjecture for groups with cyclic Sylow \(p\)-subgroups. Let \(P\) be a finite group and let \(H\) be a group of automorphisms of \(P\) such that \(p\) does not divide the order of \(H\). Then, the authors show that an \(H\)-linear separable deformation of \(kP\) can be extended to the semidirect product \(k(P\rtimes H)\). Let \(G\) be a finite group with abelian normal Sylow \(p\)-subgroup \(P\), then \(G\simeq P\rtimes H\) for a \(p'\)-subgroup \(H\) of \(G\). Then, the authors construct explicitly an \(H\)-linear separable deformation of \(kP\), hence confirming Donald-Flanigan's conjecture for \(G\).
scientific article; zbMATH DE number 897491
scientific article