Groups with a character of large degree relative to a normal subgroup. (Q403806)

Let \(N\) be a normal subgroup of a finite group \(G\) and \(\theta\) be an irreducible character of \(N\) which is fixed by the conjugation action of \(G\). Let \(\chi\) be an irreducible character of \(G\) that restricts to a multiple of \(\theta\) on \(N\). Then \(d=\chi(1)/\theta(1)\) is an integer which divides \(|G:N|\) and satisfies \(|G:N|\geq d^2\). We can thus write \(|G/N|=d(d+e)\) for a non-negative integer \(e\) and ask what can be said about \(d\) and \(G/N\) for a given \(e\).NEWLINENEWLINE In case of \(N=1\) this problem has been studied earlier by a number of authors; in particular, it has been shown that if \(e>1\), then \(d\) is bounded by a function of \(e\) which is quite well understood at this point.NEWLINENEWLINE In the paper under review the more general problem is studied under the hypothesis that \(G/N\) is solvable. It turns out that if \(e>1\) one can no longer bound \(d\) in terms of \(e\). It is shown that for \(e>0\), if \(d>(e-1)^2\), then \(e\) divides \(d\) and \(d/e+1\) is a prime power. If in addition, either \(d>e^5-e\), \((d/e,e)=1\), or \((d/e+1,e)=1\), then there exist groups \(X\), \(Y\) with \(N\leq X\trianglelefteq Y\leq G\) such that \(Y/X\) is a sharply 2-transitive group of order \((d/e)(d/e+1)\).