CHARACTERS OF π-SEPARABLE GROUPS INDUCED BY CHARACTERS OF LARGE SCHUR INDEX ROD GOW Department of Mathematics University College Belfield, Dublin 4 Ireland Let G be a finite group and let χ be an irreducible complex character of G. Let D be a complex representation of G affording χ and let m(χ) denote the Schur index of χ over the field Q of rational numbers. It is well known that m(χ) is a divisor of the degree χ(1) of χ. When the extreme case m(χ)= χ(1) occurs, D(G) is isomorphic to a finite subgroup of the multiplicative group of a division algebra. See, for example, [H, pp.548-549]. This imposes severe restrictions on the structure of D(G). For example, if p is an odd prime divisor of |D(G)|, a Sylow p-subgroup of D(G) is cyclic, and a Sylow 2-subgroup of D(G) is either cyclic or generalized quaternion. Suppose now that G is π-separable, where π is a set of primes, and let H be a Hall π-subgroup of G. Suppose also that H has an irreducible complex character θ with the property that m(θ)= θ(1). We show in this paper that, modulo certain exceptions related to the quaternion group of order 8, G has an irreducible character χ with the property that m(χ)= θ(1) and χ(1) = sθ(1), where s is a π  -number. Furthermore, the field of values Q(χ) of χ is contained in the field Q(θ). The hypothesis that m(θ)= θ(1) is somewhat restrictive and it is certainly true that it is not easy to construct non-linear characters θ with the desired property (see, for example, [I, Theorem 10.16]) but the possibility that θ is linear is included, and our main result in this case reduces to a theorem obtained by us in a previous paper, [G, Theorem 1]. Another interesting possibility is that H has an image isomorphic to a non-cyclic subgroup of the multiplicative group of the Hamiltonian quaternions 1