Characters fully ramified over a normal subgroup
Let \(H\) be a group and \(N\) a normal subgroup. Assume that \(\chi\) is an irreducible (complex) character of \(H\), and that the restriction of \(\chi\) to \(N\) is a multiple of some irreducible character of \(N\), say \(\theta\). Then \(\chi_N =e\theta\), and \(e\) is called the ramification index. It is easy to see that it always satisfies \(e^2\le \vert H:N\vert\), and when equality holds, \(\chi\) is said to be fully ramified over \(N\). This ``fully ramified case'' is the main object of study in this paper, and as an application of some of the methods, the following solvability theorem is proved. If \(H\) has an irreducible character fully ramified over a normal subgroup \(N\), and if \(p^4\) is the largest power of \(p\) dividing \(\vert H:N\vert\) for all primes corresponding to nonabelian Sylow \(p\)-subgroups of \(H/N\), then \(H/N\) is solvable.
scientific article; zbMATH DE number 3443815