An answer to a question of J.G. Thompson on some generalized characters

Abstract

In 1996 J. G. Thompson [2] proved that given any finite group G the function θp : G → Z defined by θp(g) = |{h ∈ G | 〈g, h〉p is abelian}|, where 〈g, h〉p denotes a Sylow p-subgroup of 〈g, h〉, is a generalized character. Thompson mentioned that “it seems reasonable to hope that θp is a character”. Certainly, this is the case for groups with abelian Sylow subgroups or for nilpotent groups. Unfortunately, we will show that θp does not need to be a character even for supersolvable groups with a normal Sylow p-subgroup. Let E be the extraspecial 3-group of order 33 and exponent 3. Let a be ∗Research supported by the Basque Government, the Spanish Ministerio de Ciencia y Tecnoloǵia and the University of the Basque Country