Solution of the twisting problem for skew group algebras
Let \(G\) be a finite group, \(K\) a field of prime characteristic \(p\), and \(\tau:G\to\text{Aut}(K)\) an action of \(G\) on \(K\) with kernel \(H\). If \(p\) divides the order of the subgroup \(H\), then the skew group ring \(KG\) is not semisimple. The authors [J. Pure Appl. Algebra 94, 1-15 (1994; Zbl 0809.16029)] proved that if there exists a class \(\alpha\in H^2(G,K^*)\) which ``twists'' the skew group ring \(K^\tau G\) into a semisimple crossed product \(K^\tau_\alpha G\), then \(H/O_{p'}(H)\) is an abelian group whose rank does not exceed the \(p\)-degree of \(K\) over \(K^p\). In this paper they prove that this condition is sufficient. Therefore, there is a semisimple crossed product of \(G\) over \(K\) if and only if \(G\) has a normal subgroup \(H\) such that \(G/H\) is isomorphic with a subgroup of \(\text{Aut}(K)\) and \(H/O_{p'}(H)\) is an abelian group with rank \(r\) not greater than the \(p\)-degree of \(K\) over \(K^p\).
scientific article; zbMATH DE number 820111