The permutation index of $p$-defect zero characters
Eliot T. Jacobson · Illinois Journal of Mathematics · 1988
IntroductionArtin's induction theorem asserts that any rational valued character of a finite group can be written as a rational linear combination of transitive permutation characters with cyclic stabilizer.The question of which de- nominators can occur has led to such notions as the "Artin exponent", and the "permutation index".The principal aim of this work is to study a certain divisibility property of the permutation index for p-defect zero characters.To be more specific, let G be a finite group and let X irr(G).Let sp(x) EoX where o ranges through the Galois group Gal(Q(x), Q).Then sp(x) is a rational valued character of G, hence by Artin's theorem, GIsp(x)in a Z-linear combination of permutation characters.The "permutation index" of X, denoted n(X), is the least positive integer so that n(x)sp(x) is such a combination.The p-Defect Zero Conjecture.Let G be a finite group, and let p be a prime.Let X irr(G) satisfy p IGI/x(1) (X has p-defect zero).Then p n(x).This natural conjecture arose by consideration of Solomon's similar result for the Schur index [5].Gluck's work [1] is applicable to some extent; his ideas led to Proposition 4. We discuss the significance of his methods in Theorem F. Parks' work [4] is also helpful in understanding the problem, however the configuration that makes his proof go through is nowhere in sight.Thus a proof for even solvable G seems quite remote at present.We present here several cases in which the conjecture holds.We use these cases as evidence towards the general result, hoping to inspire greater interest in this and similar problems.The author expresses his gratitude to professors Alan Parks and Tom Wolf for their helpful discussions.