Extremely strong Shoda pairs with \texttt{GAP}

TL;DR

This paper introduces algorithms for computing strong and extremely strong Shoda pairs of finite groups, which help determine primitive central idempotents of their rational group algebras, enhancing understanding of their algebraic structure.

Contribution

It presents new algorithms for identifying all strong and extremely strong Shoda pairs and primitive central idempotents of finite groups, including a method to check if a group is normally monomial.

Findings

Algorithms successfully compute all strong and extremely strong Shoda pairs.

The methods enable explicit determination of primitive central idempotents.

A new algorithm verifies if a group is normally monomial.

Abstract

We provide algorithms to compute a complete irredundant set of extremely strong Shoda pairs of a finite group $G$ and the set of the primitive central idempotents of the rational group algebra $\mathbb{Q}[G]$ realized by them. These algorithms are also extended to write new algorithms for computing a complete irredundant set of strong Shoda pairs of $G$ and the set of the primitive central idempotents of $\mathbb{Q}[G]$ realized by them. Another algorithm to check whether a finite group $G$ is normally monomial or not is also described.