Group Multiplication Explorer (2D)

Group Family

Order n 6

Generator g 1

Mode

Live Readout

|G| (order)

Selected element order

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Click an element, then another

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In multiply mode, click element A, then click B: the graph looks up the real group product A·B from the group's own multiplication law (not an arbitrary pairing). In subgroup mode, click any element h to highlight the cyclic subgroup ⟨h⟩ = {e, h, h², …} it generates.

How the Cayley graph is built

Nodes are the group's actual elements. A thin edge from x to x·g is drawn for each generator g — that edge set is exactly the definition of a Cayley graph. For ℤₙ, elements are 0..n−1 under addition mod n. For the dihedral group Dₙ (symmetries of a regular n-gon), elements are rⁱ (rotations, inner ring) and s·rⁱ (reflections, outer ring), multiplied with the real relation s·r·s = r⁻¹, so (rᵃsᵇ)(rᶜsᵈ) = r^(a + c·(−1)ᵇ) s^(b+d).

Rotation-generator edges are drawn as arrows; reflection edges (order 2) as plain lines, since s·s = e makes them symmetric.
Every subgroup shown here is a genuine cyclic subgroup ⟨h⟩, whose order always divides |G| — Lagrange's theorem, visible directly in how many nodes light up.