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

2024/01/13 by Gurmeet K. Bakshi, Bakshi, Gurmeet K., Jyoti Garg +3 · 1 citation
Mathematics · #16K20 #16S35 #16U60 #17C27 #20C05 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2401.07101

openalex publication_date 2024/01/13 · openalex created_date 2024/01/18 · openalex updated_date 2026/07/28

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 ℚG for G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in ℚG, we describe a subgroup of finite index in the group of units U(ℤG) of the integral group ring ℤG 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 is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.

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