Group characters, permutation actions and sharpness

AbstractWe extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X. Further consequences for ordinary group characters are obtained by re-examining Blichfeldt’s work and we provide alternative definitions of sharpness. Conjectures are given related to the set of values of a permutation character, and it is proved that for a faithful transitive G-set X certain polynomials (in the Burnside ring) evaluated at X necessarily give G-sets