Cayley: Group Theory Explorer

Unlock the world of abstract algebra with Cayley, an interactive educational tool designed for students, educators, and mathematics enthusiasts. Cayley provides a visual and computational approach to exploring finite group structures, with a focus on symmetric groups and dihedral symmetries.
Whether you are learning about group theory for the first time or looking for a way to visualize complex algebraic concepts, this app simplifies the study of discrete mathematics.
Key Features:
Permutation Analysis Define permutations in one-line notation and instantly generate their disjoint cycle decomposition. Analyze essential properties including element order and parity (even or odd signatures).
Group Composition Tables Generate and explore Cayley tables for symmetric groups ($S_n$). The interactive grids use color-coding to highlight the identity element and group patterns, making it easier to understand concepts like closure, associativity, and inverses.
Dihedral Symmetry Explorer Visualize the symmetries of regular polygons with the Dihedral group ($D_n$) tool. Interact with triangles, squares, pentagons, and hexagons through rotations and reflections to see how physical transformations correspond to group elements.
Element Order Calculator Input any permutation to view a step-by-step breakdown of its powers. This feature helps you visualize how an element generates a cyclic subgroup and the exact number of steps required to reach the identity.
Educational Reference Access built-in mathematical notes and quick definitions. Learn about fundamental principles such as Cayley’s Theorem, cycle notation, and the relationship between cycle lengths and element orders.
Cayley is built to turn abstract mathematical theories into a tangible, interactive experience, helping you discover the beauty of symmetry and the structure of finite groups.