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Every way to visualize a group's structure

Four different pictures of the same object -- each throws away different information to make something else easy to see.

Reach for a Cayley table to verify the group axioms directly or spot commutativity at a glance (a symmetric table means abelian). Reach for a subgroup lattice when the question is about containment. Reach for a Cayley graph when the question is about generators and reachability rather than the full multiplication. And reach for the group acting directly on a real object when the question is why a group behaves the way it does, not just that it does.

See every product at once

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A square table listing every operation's result, rows and columns both indexed by the set's elements.

Example

D_3's 6x6 table is not symmetric across its diagonal, confirming D_3 is non-abelian before a single element pair is checked by hand.

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A square grid of every product -- closure, identity, and inverses all read off directly; a symmetric table (across the diagonal) means the group is abelian.

See every subgroup at once, by containment

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A Hasse diagram showing every subgroup of a group, arranged by rank, with an edge between H and K exactly when H is covered by K.

Example

D_4's lattice shows three subgroups directly below D_4 itself and five directly above \{e\}, with two of those five unrelated to each other by containment.

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One box per subgroup, arranged by rank, with an edge whenever one subgroup is covered by another -- containment becomes a matter of following edges downward.

See only the generators, and how far they reach

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A graph built from a group G and a generating set S: one vertex per element of G, and one directed edge x \to xs for every vertex x and every s \in S.

Example

\operatorname{Cay}(\mathbb{Z}_6,\{1\}) is a hexagon -- one arrow out of every vertex, all pointing the same way around.

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Throws away every product except the ones built from a chosen generating set -- one vertex per element, one arrow per generator, so structure shows up as a shape instead of a grid.

Watch the group act directly on a physical object

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The group of all 2n symmetries of a regular n-gon -- n rotations and n reflections -- under composition.

Example

On a square, Apply-s then Apply-r lands on a different symmetry than Apply-r then Apply-s -- the same rs\ne sr fact D_4's Cayley table would show as an asymmetric entry pair.

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This is the object the other three pictures describe abstractly -- applying r and s to a real polygon and comparing rs against sr shows non-abelian behavior directly, before any table is even built.

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