INVESTIGATION INTO FIVE KEY PROPERTIES OF THE ORDER PRODUCT PRIME GRAPHS OF ALTERNATING GROUPS OF DEGREE FIVE AND SIX

Abstract

Graph theory has become a key tool in group theory research, enabling the investigation of algebraic structures through graph properties. This has led to the evolution of diverse group graph definitions. This study explored the connection between alternating groups and their order product prime graphs. We constructed the order product prime graphs for alternating groups of degree 5 and 6, analyzing properties like regularity, completeness, connectivity, girth and diameter. Using 'Group Algorithm Programming Software' we generated group elements and built the graphs. Our findings revealed that the order product prime graphs of alternating groups of degrees 5 and 6 are connected, irregular, incomplete, and non-planar, with a consistent topology featuring a girth of 3 and diameter of 2, indicating a stable and well-connected structure.