Rational Group Algebras of Camina $p$-groups

TL;DR

This work delivers a concrete, combinatorial description of the Wedderburn decomposition of the rational group algebra $\mathbb{Q}G$ for Camina $p$-groups, extending prior results for abelian and VZ $p$-groups to a broader nonabelian Camina context. It provides explicit decomposition formulas for nilpotency classes $r=2$ and $r=3$, distinguishing cases by the prime $p$ and yielding components such as $\mathbb{Q}(\zeta_p)$, $M_{n}(\mathbb{Q}(\zeta_p))$, and even $\mathbb{H}(\mathbb{Q})$ in the $p=2$ quaternion setting. A complete description of primitive central idempotents is achieved via $e_{\mathbb{Q}}(\chi)$ and the $\epsilon(G,N)$ framework, with class-2 and class-3 analyses detailing how $Z(G)$ and $G'$ influence the simple components. The results also show that isoclinic Camina $p$-groups share isomorphic rational group algebras, and together these findings enhance understanding of how Camina group structure governs the semisimple rational representation theory.

Abstract

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of Camina $p$-groups, where $p$ is a prime. We also provide a complete set of primitive central idempotents of rational group algebras of these groups.

Theorems & Definitions (26)

• Theorem 1
• Corollary 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9: Perlis-Walker Theorem
• Lemma 10