The minimal degree of a non-linear irreducible character of a solvable group (Q1185965)

We are concerned here with the question: What is the smallest degree of a non-linear complex irreducible character of a finite solvable group? If \(G\) is such a group, and \(\mu\) is an irreducible non-linear character of minimal degree of \(G\), chosen to have as large a kernel as possible, we can factor out the kernel of \(\mu\), and then a reasonably complete answer to our question is provided by: Theorem 1. Let \(G\) be a finite non- Abelian solvable group, and let \(\mu\) be a non-linear complex irreducible character of \(G\) of least degree. Assume that whenever \(H\) is a non- identity normal subgroup of \(G\), all non-linear irreducible characters of \(G/H\) have degree strictly larger than \(\mu(1)\). Then one of the following occurs: (a) \(G\) is a \(p\)-group of class 2 for some prime \(p\), \(| G'| = p\), \(Z(G)\) is cyclic, and \(\mu(1) = [G: Z(G)]^{1/2}\). (b) \(G\) is a Frobenius group of the form \(PA\), where \(A\) is a cyclic complement acting irreducibly on the elementary Abelian \(p\)-group \(P\), and \(\mu(1) = | A|\). (c) \(G = Z(G)H\), where, for some prime \(p\), \(Z(G)\) is a cyclic \(p\)-group, \(O_ p(H) = G'\) is a non-Abelian special \(p\)-group, \(G\) has a cyclic Hall \(p'\)-subgroup \(A\) such that \(AG'/G''\) is a Frobenius group with complement \(AG''/G''\), and setting \(A_ 0 = C_ A(G'')\), we have \(\mu(1) = [A: A_ 0](| A_ 0| - 1)\). Furthermore, \(| A_ 0| - 1\) is a power of \(p\), say \(p^ e\), and there is a subgroup \(T\) of \(G''\) such that \(G'/T\) is a direct product of extra-special groups of order \(2^{2e + 1}\), each of which is invariant under the action of \(A_ 0 T/T\), with the latter group acting trivially on the centre and irreducibly and faithfully on the Frattini factor group.