Groups in Motion

Abstract Algebra · Axiom Academy

Discover how groups act on sets through rotating polygons, permuting vertices, and preserving patterns A square can be rotated by 90°, 180°, 270°, or 360° (back to start). Click the buttons to see how each rotation transforms the square. Watch how the colored vertices move! Each rotation doesn't just spin the square—it rearranges (permutes) the vertices. A 90° rotation moves vertex A to where B was, B to where C was, and so on. What happens when we perform two rotations in sequence? Click the button below to see that rotating 90° twice gives the same result as rotating 180° once! Symmetry and Coloring Patterns Some colorings have special symmetry properties. Apply different patterns and test which rotations preserve them. This introduces the concept of stabilizers! A group G acts on a set X when each group element g gives a bijection of X. The rotation group acts on the square's vertices, giving permutations that compose according to the group operation. Each rotation is a bijection (one-to-one and onto). When we compose two rotations, the resulting bijection corresponds to the group product: r₉₀ ∘ r₉₀ = r₁₈₀. The composition of functions mirrors group multiplication! For any vertex, its orbit is all positions it can reach (all four corners). The stabilizer of a coloring pattern is the subgroup of rotations preserving it. These concepts are fundamental to understanding group actions!

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