Build stunning diagrams that reveal a group's entire structure at a glance
A Cayley diagram is a directed graph that encodes the entire structure of a group at a glance. Each node represents a group element and each colored arrow represents multiplication by a generator. Follow the arrows from the identity to discover every element.
Named after Arthur Cayley, these diagrams reveal symmetry, subgroup structure, and the "shape" of a group in a way that multiplication tables cannot.
Watch the diagram build itself node-by-node via BFS from the identity. Each step discovers a new element by multiplying an existing element by a generator. Colored arrows show which generator was used.
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Key insight: The BFS construction mirrors how generators create the entire group. The number of steps to reach an element from the identity is its "word length" in the generators.
Browse pre-built Cayley diagrams for various groups. Click any node to highlight its outgoing edges and see the element's label and order.
Click a node to highlight its outgoing edges.
Order: 6
Key insight: Different groups have strikingly different diagram shapes. Cyclic groups form circles, dihedral groups look like polygons with spokes, and the quaternion group Q8 has a distinctive 3D-like structure.
Type a word using the generators and watch it trace a path through the Cayley diagram. Each letter follows the corresponding colored arrow. Different words can lead to the same element -- these are the group's relations.
Type a word using generators r and s for S3, then trace the path through the Cayley diagram.
Key insight: A group is completely determined by its generators and relations. For S3: r^3 = e, s^2 = e, srs = r^-1. Try words like "rrr" or "ss" to see them loop back to the identity.