Thompsonʼs generalized characters and permutation characters

Given a property P of groups and a finite group G (not necessarily having this property) J.G. Thompson (1996) [5] defined an associated counting function χP on G. For certain properties P he then establishes that χP is a generalized character of G. We prove here that, under mild conditions on P, these functions are not only generalized characters but in fact lie in the permutation character ring of G.