Visualization of Cyclic, Dihedral, and Symmetric Groups

Group theory plays an important role in mathematics, providing a framework to understand symmetries and structures across various fields. This study explores visualization techniques for cyclic, dihedral, and symmetric groups. Visual tools such as Circle Representations and Cayley graphs are employed to illustrate and analyze group properties, including element orders, inverses, group operations, and subgroup structures. In the case of cyclic groups, in addition to Cayley graphs, the circle representation was also used to geometrically model the group structure. For more complex groups such as symmetric groups and dihedral groups , Cayley graphs were studied to understand how group generators relate to group elements and to visualize their structural symmetries and subgroup formations. While Cayley graphs for cyclic, dihedral, and symmetric groups have been previously visualized, for instance, through platforms like Group Explorer (Carter, n.d.), there remains a lack of algorithms and tools that enable users to perform group operations, compute element orders and inverses, and identify subgroup structures directly from these visualizations. For cyclic, dihedral, and symmetric groups , algorithms were developed to automate tasks such as determining the order and inverse of elements, performing group operations, and identifying subgroup elements using Cayley graphs. These algorithms were realized through Python-based web applications developed with the Flask framework. The applications allow users to interactively explore group visualizations and perform computations related to group properties, enhancing both understanding and usability for learners and researchers. The results demonstrate that visual representations, when supported by algorithmic analysis, provide powerful tools for grasping abstract group theoretic concepts. The developed applications successfully link theoretical foundations with computational exploration, offering an effective means for learning, teaching, and further research in abstract algebra. This study highlights how visualization bridges intuition and formalism in group theory and contributes to educational tools and computational mathematics.