Characters of $$ \pi $$ -separable groups induced by characters of large Schur index

Abstract

Let G be a finite \( \pi \) -separable group, where \( \pi \) is a set of primes, and let H be a Hall \( \pi \) -subgroup of G. Suppose that H has an irreducible complex character \( \theta \) with the property that \( m(\theta) = \theta(1) \) , where \( m(\theta) \) denotes the Schur index of \( \theta \) over the rational field. We show in this paper that, modulo certain exceptions related to the group SL2 (3) of order 24, G has an irreducible character \( \chi \) with the property that \( m(\chi) = \theta(1) \) and \( \chi(1) = s\theta(1) \) , where s is a \( \pi' \) -number. The character \( \chi \) is distinguished by the property that it is the unique constituent of \( \theta^G \) whose restriction to a Hall \( \pi' \) -subgroup M, say, of G contains the principal character 1M with non-zero multiplicity. We also include a result about characters of 2-solvable groups induced by irreducible characters of degree 2 of a Sylow 2-subgroup.