Asia Mathematika, Vol 9, issue 3, pages: 47--59.

Determination of the transitivity and primitivity of dihedral groups of prime degrees that are not p-groups using numerical approach

Ben O. Johnson
Department of Mathematics, Federal University, Wukari. Wukari, Nigeria.
Adagba T. Titus
Department of Mathematics, Federal University, Wukari. Wukari, Nigeria.

Received: 12 Oct 2025 | Accepted: 15 Dec 2025 | Final Version: 31 Dec 2025

Abstract

In this study, we carried out further study on the transitive and primitive nature of dihedral group of prime degrees that are not p-groups by numerical approach. Transitivity and primitivity are two pivotal properties that provide deeper insights into group structures. Primitive groups represent the building blocks of all finite groups, akin to prime numbers in number theory. Transitivity, on the other hand, reflects a group's ability to act uniformly on a set, highlighting its symmetrical properties. A group $G$ acting on a set $\Omega$ is said to be transitive on $\Omega$ if it has one orbit and so $\alpha^{G}=\Omega$ for all $\alpha \in \Omega$. Equivalently, $G$ is transitive if for every pair of point $\alpha, \beta \in \Omega$ there exists $g \in G$ such that $\alpha^{g}=\beta$. A permutation group $G$ acting on a non empty set $\Omega$ is called primitive if $G$ acts transitively on $\Omega$ and $G$ preserves no non trivial partition of $\Omega$. In other words, a group $G$ is said to be primitive on a set $\Omega$ if the only sets of imprimitivity are the trivial ones otherwise $G$ is imprimitive on $\Omega$. In this work we generated some dihedral groups of prime degrees that are not p -groups and used computational tools, including GAP (Groups, Algorithms, and Programming) coupled with maximality theorem to analyze their structures and action properties and discuss their transitive and primitive nature. The findings contribute to a deeper understanding of finite permutation groups, offering new insights into their classification and properties. This study not only enriches the theoretical framework of abstract algebra but also provides practical applications in areas such as cryptography, chemistry, and computational group theory.

Keywords:
Transitive groups, primitive groups, prime degree, p-groups, numerical approach, group theory, GAP

1. Introduction

Group theory plays a pivotal role in many areas of mathematics, especially where symmetry is a key consideration. Symmetry in any object is inherently tied to group theory, making it difficult to discuss symmetry without referencing this mathematical framework. As one of the foundational branches of abstract algebra, group theory seeks to classify groups up to isomorphism. This means that, for any given group, it should be possible to identify a corresponding known group via an isomorphism... \end{equation}

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