Groups with three real valued irreducible characters

We obtain strong restrictions on the structure of a Sylow 2-subgroup of a group with at most three real valued irreducible characters. This extends results of Iwasaki, who studied groups with at most two real valued irreducible characters. This research was partially supported by the Spanish Ministerio de Educación y Ciencia, MTM2004-06067-C02-01 and MTM2004-04665, and the FEDER. The first author was also supported by the Programa Ramón y Cajal and the Generalitat Valenciana.