Characters fully ramified over a normal subgroup
type: Item
instance of: scholarly article
title: Characters fully ramified over a normal subgroup
zbMATH Open document ID: 0283.20003
author: Stephen M. jun. Gagola
published in: Pacific Journal of Mathematics
publication date: 1975-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 3443815
MaRDI profile type: Publication
full work available at URL: pjm.1974.55.107
OpenAlex ID: W2031504552
DOI: 10.2140/PJM.1974.55.107
description: scientific article; zbMATH DE number 3443815