The Rationality of Sylow 2-Subgroups of Solvable Q_1-Groups

A finite group whose irreducible complex non-linear characters are rational is called a Q1-group. In this paper,we studyQ1-groups via their Sylow2-subgroups. In particular, we prove that if G is a solvable Q1-group that is not a rational group, then its Sylow 2- subgroups areQ1-groups. To prove this,we obtain information aboutQ1-groups where a Sylow2-subgroup is contained in the vanishing off subgroup where the vanishing-off subgroup is a proper subgroup.We also obtain results about Q1-groups where a Sylow 2-subgroup is not contained in the vanishing off subgroup. In addition, we prove that Q1-groups whose Sylow 2-subgroups are dihedral and generalized quaternion groups have that all of the irreducible characters are rational groups.