Rational group algebras of generalized strongly monomial groups: Primitive idempotents and units

Abstract:

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra $\mathbb {Q}G$ for $G$ a finite generalized strongly monomial group. For the same groups with no exceptional simple components in $\mathbb {Q}G$, we describe a subgroup of finite index in the group of units $\mathcal {U}(\mathbb {Z}G)$ of the integral group ring $\mathbb {Z}G$ that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable.