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Rational group algebras of generalized strongly monomial groups: primitive idempotents and units
Rings and Algebras · Submitted 13 Jan 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.88 | 0.96 | 0.59 | asserted 1.00 |
| s1 | 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. | contribution 0.85 | 0.95 | 0.20 | asserted 1.00 |
| s2 | We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. | contribution 0.56 | 0.94 | 0.05 | asserted 0.99 |
| s3 | 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. | result 0.77 | 0.88 | 0.09 | asserted 1.00 |
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Original 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 is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.
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