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On Matrix Representations of Groups of Order $p^5$ over $\mathbb{Q}$

Ram Karan Choudhary, Sunil Kumar Prajapati

TL;DR

This paper addresses the problem of constructing all inequivalent irreducible rational matrix representations of groups of order $p^5$ over $\mathbb{Q}$ for odd primes $p$, and derives combinatorial descriptions for the Wedderburn decomposition of their rational group algebras. It employs the Ram algorithm via required pairs $(H, \psi)$, organized through James’ $10$ isoclinic families $\Phi_i$, to produce explicit rational representations for all such groups and to describe their simple components in $\mathbb{Q}G$. The results cover all families $\Phi_2$–$\Phi_{10}$ with detailed constructions, lifting from suitable quotients and using Galois conjugacy to determine fields of realization, along with concrete examples and computational verification. The work provides both theoretical classifications and practical Magma tools for explicit realization of rational representations and Wedderburn decompositions, with potential applications to the study of rational group algebras of $p$-groups.

Abstract

In this article, we determine all inequivalent irreducible rational matrix representations of groups of order $p^5$, where $p$ is an odd prime. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these $p$-groups, using results from their rational representations.

On Matrix Representations of Groups of Order $p^5$ over $\mathbb{Q}$

TL;DR

This paper addresses the problem of constructing all inequivalent irreducible rational matrix representations of groups of order over for odd primes , and derives combinatorial descriptions for the Wedderburn decomposition of their rational group algebras. It employs the Ram algorithm via required pairs , organized through James’ isoclinic families , to produce explicit rational representations for all such groups and to describe their simple components in . The results cover all families – with detailed constructions, lifting from suitable quotients and using Galois conjugacy to determine fields of realization, along with concrete examples and computational verification. The work provides both theoretical classifications and practical Magma tools for explicit realization of rational representations and Wedderburn decompositions, with potential applications to the study of rational group algebras of -groups.

Abstract

In this article, we determine all inequivalent irreducible rational matrix representations of groups of order , where is an odd prime. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these -groups, using results from their rational representations.

Paper Structure

This paper contains 12 sections, 33 theorems, 43 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Rational matrix representations
  5. Groups belonging to $\Phi_2$ and $\Phi_5$
  6. Groups belonging to $\Phi_4$ and $\Phi_6$
  7. Groups belonging to $\Phi_3$ and $\Phi_9$
  8. Groups belonging to $\Phi_7$, $\Phi_8$ and $\Phi_{10}$
  9. Examples
  10. Rational group algebras
  11. Computational verification
  12. Acknowledgments

Key Result

Lemma 1

SKP Let $(G, Z(G))$ be a Camina pair. Then there exists a bijection between the sets $\textnormal{Irr}(G | Z(G))$ and $\textnormal{Irr}(Z(G)) \setminus \{1_{Z(G)}\}$, where $1_{Z(G)}$ is the trivial character of $Z(G)$. For $1_{Z(G)} \neq \mu \in \textnormal{Irr}(Z(G))$, the corresponding $\chi_\mu

Theorems & Definitions (64)

  • Definition 3.0.1
  • Lemma 1
  • Lemma 2
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • proof
  • Theorem 8
  • proof
  • ...and 54 more