Modular Frobenius groups
Let \(G\) be a finite group and let \(N\) be a proper non-trivial normal subgroup of \(G\). Furthermore, let \(K\) be a splitting field of characteristic \(p\) for \(N\). Then \(G\) is called a (\(p\)-)modular Frobenius group if for every non-trivial irreducible \(KN\)-module \(V\) the induced module \(V^G\) is irreducible. Note that modular Frobenius groups in characteristic zero are just Frobenius groups. In particular, the Frobenius kernel \(N\) is nilpotent, by a result of Thompson. In the paper under review modular Frobenius groups are studied for a prime \(p\). As a main result the authors prove that \(N\) is solvable if \(p\) is odd while the non-cyclic composition factors of \(N\) are isomorphic to \(\text{PSL}(2,3^{2^t})\), \(t\geq 1\), if \(p=2\). The proof depends on the classification of finite simple groups. For any \(t\geq 1\) examples are given in which the kernel \(N\) involves \(\text{PSL}(2,3^{2^t})\). The authors also investigate the structure of solvable modular Frobenius groups.
scientific article; zbMATH DE number 980704
scientific article