On the Order Product Prime Graphs for Alternating Groups of Degree Three and Four

Abstract

Graph-based approaches have emerged as a key area of research in group theory, with a growing focus on leveraging graph properties to investigate algebraic structures. Research on the graphical structure of finite groups has led to the development of various definitions of group graphs over time. This study explored the relationship between alternating groups and their respective order product prime graphs. The research constructed the order product prime graphs for alternating groups of degree three and four. Regularity, completeness, connectivity, girth and diameter of each graph constructed were also checked. The study utilized the 'Group Algorithm Programming Software' to generate the elements of the alternating groups investigated, facilitating the construction of their order product prime graphs. The findings revealed that the order product prime graphs of alternating groups of degrees three and four are connected. The study discovered that the graph of degree three has the distinctive properties of completeness, regularity, and planarity, a trait that sets it apart from the graphs of degree four. The graph of degree three has a girth of 3 and a diameter of 1, indicating a highly connected and compact structure. Notably, the graphs of degree four, exhibits a girths of 3 and diameters of 2, indicating a stable and well-connected structure.