Asia Mathematika, Vol 9, issue 3, pages: 60--73.

A study on algebraic properties of permutation group using numerical approach

Ben O. Johnson
Department of Mathematics, Federal University, Wukari. Wukari, Nigeria.
Mtema Zakaa
Department of Mathematics, Federal University, Wukari. Wukari, Nigeria.

Received: 20 Oct 2025 | Accepted: 15 Dec 2025 | Final Version: 31 Dec 2025

Abstract

In this study, we carried out further study on permutation group of degrees n, where n is a positive integer. Commutativity and Transitivity are two pivotal properties that provide deeper insights into group structures. Commutativity in Group Theory refers to the property where the order of elements in a group operation does not affect the result. A group (G,*) is said to be commutative or abelian if: \(a*b=b*a, \forall a,b \in G\). 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 \(\alpha^{g}=\beta\). In this work we generated some symmetric groups of degree n as a good example of permutation group and used computational tools, including Groups, Algorithms, and Programming (GAP) to analyze their structures and action properties and discuss their commutativity and transitivity. It was found that symmetric groups of degrees n < 3 are commutative and transitive while non commutative but transitive otherwise. These 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 and computational group theory.

Keywords:
Commutative groups, transitive groups, permutation group, p-groups, numerical approach, group theory, GAP

1. Introduction

Group theory is a fundamental area in abstract algebra, essential for studying mathematical structures and symmetry in various systems. It provides the framework to explore sets equipped with operations that satisfy particular axioms, including closure, associativity, identity, and invertibility... \end{equation}

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