Characters of \(\pi\)-separable groups induced by characters of large Schur index. (Q1881014)

Let \(D\) be a complex representation of a group \(G\) affording the character \(\chi\) and let \(m(\chi)\) be the Schur index of \(\chi\) over the field \(\mathbb{Q}\) of the rational numbers. It holds that \(m(\chi)\) divides \(\chi(1)\), the degree of \(\chi\). In the extreme case \(m(\chi)=\chi(1)\) one gets that for an odd prime \(p\) dividing \(|D(G)|\), a Sylow \(p\)-sugroup of \(D(G)\) is cyclic whereas a Sylow 2-subgroup of \(D(G)\) is either cyclic or generalized quaternion. In this paper, a lot of lemmas and theorems around this theme are proved. We mention a few ones. Lemma 5: Let \(N\) be a normal Hall subgroup of \(G\). Suppose \(N\) has a faithful irreducible complex character \(\psi\) satisfying \(m(\psi)=\psi(1)\). Suppose also that \(N\geq C_G(N)\). Then the inertia subgroup \(I\) of \(\psi\) in \(G\) equals \(N\) or \(N\) has index 3 in \(I\). The very last case might only occur when \(N\) is isomorphic to a quaternion group of order eight. We give a part of the main result of this paper: Part of Theorem 1. Let \(G\) be \(\pi\)-separable, \(H\) a Hall-\(\pi\)-subgroup and \(M\) a Hall-\(\pi'\)-subgroup. Suppose \(H\) has an irreducible complex character \(\vartheta\) satisfying \(m(\vartheta)=\vartheta(1)\). Suppose also that \(G\) does not involve \(\text{SL}(2,3)\). Then \(\vartheta^G\) contains a unique irreducible character \(\chi\) for which \((\chi_M,1_M)>0\). As such, \((\chi_M,1_M)=\vartheta(1)\), \(\mathbb{Q}(\chi)\leq\mathbb{Q}(\vartheta)\) and \(m(\chi)=\vartheta(1)\). Moreover, there is a subgroup \(U\) of \(G\) that contains \(H\) and an irreducible character \(\varphi\) of \(U\) with \(\varphi_H=\vartheta\) and \(\varphi^G=\chi\).