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A Combinatorial Technique for the Wedderburn Decomposition of Rational Group Algebras of Nested GVZ $p$-groups

Ram Karan Choudhary, Sunil Kumar Prajapati

TL;DR

The paper develops a comprehensive combinatorial framework for the Wedderburn decomposition of rational group algebras of nested GVZ $p$-groups with odd prime $p$. It first proves a general decomposition formula for nested GVZ $p$-groups, then specialized results for two-generator $p$-groups of class $2$, including explicit expressions in terms of center-quotient data and cyclic-subgroup counts. It further exhibits two families of nested GVZ $p$-groups with arbitrarily large nilpotency class and derives their decompositions, and proves that GVZ/nested GVZ properties are preserved under isoclinism, enabling a classification of all nested GVZ $p$-groups of order at most $p^5$ and a detailed account of primitive central idempotents. Collectively, these results connect the representation-theoretic structure of $ olinebreak \\mathbb{Q}G$ with concrete combinatorial invariants of the group, aiding explicit computations and software implementations in Wedderga and related tools.

Abstract

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of nested GVZ $p$-groups, where $p$ is an odd prime. Using this formula, we derive an explicit combinatorial expression for the Wedderburn decomposition of rational group algebras of all two-generator $p$-groups of class $2$. Additionally, we provide explicit combinatorial formulas for the Wedderburn decomposition of rational group algebras of certain families of nested GVZ $p$-groups with arbitrarily large nilpotency class. We also classify all nested GVZ $p$-groups of order at most $p^5$ and compute the Wedderburn decomposition of their rational group algebras. Finally, we determine a complete set of primitive central idempotents for the rational group algebras of nested GVZ $p$-groups.

A Combinatorial Technique for the Wedderburn Decomposition of Rational Group Algebras of Nested GVZ $p$-groups

TL;DR

The paper develops a comprehensive combinatorial framework for the Wedderburn decomposition of rational group algebras of nested GVZ -groups with odd prime . It first proves a general decomposition formula for nested GVZ -groups, then specialized results for two-generator -groups of class , including explicit expressions in terms of center-quotient data and cyclic-subgroup counts. It further exhibits two families of nested GVZ -groups with arbitrarily large nilpotency class and derives their decompositions, and proves that GVZ/nested GVZ properties are preserved under isoclinism, enabling a classification of all nested GVZ -groups of order at most and a detailed account of primitive central idempotents. Collectively, these results connect the representation-theoretic structure of with concrete combinatorial invariants of the group, aiding explicit computations and software implementations in Wedderga and related tools.

Abstract

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of nested GVZ -groups, where is an odd prime. Using this formula, we derive an explicit combinatorial expression for the Wedderburn decomposition of rational group algebras of all two-generator -groups of class . Additionally, we provide explicit combinatorial formulas for the Wedderburn decomposition of rational group algebras of certain families of nested GVZ -groups with arbitrarily large nilpotency class. We also classify all nested GVZ -groups of order at most and compute the Wedderburn decomposition of their rational group algebras. Finally, we determine a complete set of primitive central idempotents for the rational group algebras of nested GVZ -groups.

Paper Structure

This paper contains 12 sections, 27 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Preliminaries
  5. Rational group algebra
  6. Two-generator $p$-groups of nilpotency class $2$
  7. An explicit formula for $\mathbb{Q}G_{(\alpha,\beta,\gamma;\rho,\sigma)}$ with $(\alpha,\beta,\gamma;\rho,\sigma) \in \tau_{n_5}$
  8. Nested GVZ $p$-groups of arbitrarily large nilpotency class
  9. Nested GVZ $p$-groups of arbitrary large nilpotency class and exponent $p$
  10. Nested GVZ $p$-groups of arbitrary large nilpotency class and arbitrary large exponent
  11. Isoclinic nested GVZ $p$-groups
  12. Primitive central idempotents

Key Result

Theorem 1

PW Let $G$ be a finite abelian group of exponent $m$. Then the Wedderburn decomposition of $\mathbb{Q}G$ is given by where $a_d$ is equal to the number of cyclic subgroups of $G$ of order $d$.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 33 more