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

Gurmeet K. Bakshi, Jyoti Garg, Gabriela Olteanu

TL;DR

This work develops a constructive framework to compute a complete set of orthogonal primitive idempotents in a simple component of the rational group algebra $\mathbb{Q}G$ with Schur index 1 for finite generalized strongly monomial groups, and to describe a finite-index subgroup of $\mathcal{U}(\mathbb{Z}G)$ when no exceptional components occur. The approach combines generalized strong Shoda pairs with explicit matrix-unit and center decompositions, yielding practical methods to obtain primitive idempotents, matrix units, and generators for large subgroups of the unit group. It is illustrated through a detailed example $G = P \rtimes D_{2^n}$ and extended to Frobenius groups of odd order with cyclic complements, showing broad applicability of the theory to classical group-ring questions. The results deepen understanding of the Wedderburn structure and unit groups in this broad class of groups, enabling explicit computations of primitive idempotents and units in concrete cases.

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.

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

TL;DR

This work develops a constructive framework to compute a complete set of orthogonal primitive idempotents in a simple component of the rational group algebra with Schur index 1 for finite generalized strongly monomial groups, and to describe a finite-index subgroup of when no exceptional components occur. The approach combines generalized strong Shoda pairs with explicit matrix-unit and center decompositions, yielding practical methods to obtain primitive idempotents, matrix units, and generators for large subgroups of the unit group. It is illustrated through a detailed example and extended to Frobenius groups of odd order with cyclic complements, showing broad applicability of the theory to classical group-ring questions. The results deepen understanding of the Wedderburn structure and unit groups in this broad class of groups, enabling explicit computations of primitive idempotents and units in concrete cases.

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 for a finite generalized strongly monomial group. For the same groups with no exceptional simple components in , we describe a subgroup of finite index in the group of units of the integral group ring 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.
Paper Structure (18 sections, 18 theorems, 97 equations)

This paper contains 18 sections, 18 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Primitive Idempotents
  4. Units
  5. Example 1: $G = P \rtimes D_{2^n}$
  6. The presentation of the group
  7. Wedderburn decomposition
  8. Primitive Idempotents
  9. $(H,K)\in \left\{ (G,G), (G,\langle P,a^2,ab \rangle), (G,\langle P,a \rangle), (G,\langle P,a^2,b \rangle)\right\}$.
  10. $(H,K)$ with $H=\langle P,a \rangle$ and $K = \langle P,a^{2^i} \rangle,~2 \leq i \leq n-1$.
  11. $(H,K)$ with $H=\langle P,b \rangle$ and $K \lhd H$, $H/K$ cyclic, $K \neq P$.
  12. $(H,K)$ with $H=\langle P,ba \rangle$ and $K\lhd H$, $H/K$ cyclic, $K \neq P$.
  13. $(H,K)$ with $H=P$ and $K \unlhd H$, $H/K$ cyclic, $K\in Z$.
  14. $(H,K)$ with $H=P$ and $K \unlhd H$, $H/K$ cyclic, $K \notin Y$ and $K \notin Z$.
  15. $(H,K)$ with $H=\langle x,y_{1},\ldots,y_{r},a \rangle$ and $K=\langle y_{1},\ldots,y_{r},a^{2^j}\rangle$, for $0 \leq j \leq n-1$.
  16. ...and 3 more sections

Key Result

Theorem 2.4

(BK19, Theorem 3) Let $(H,K)$ be a generalized strong Shoda pair of $G$ and $\lambda$ a linear character of $H$ with kernel $K$. Let $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ be a strong inductive chain from $H$ to $G$. Then where $C_{i},~\sigma_{H_{i}},~\tau_{H_{i}},~k_{i}$ are as defined above.

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.4
  • Theorem 2.6
  • Remark 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 15 more