The Group Zoo

Abstract Algebra · Axiom Academy

Explore the fascinating variety of mathematical groups through interactive exhibits. Discover patterns, symmetries, and surprising connections! Just like a zoo has different species of animals, mathematics has different species of groups . Each exhibit showcases a different type of group. Click any exhibit to begin your tour! Number Groups: Clock Arithmetic Meet ℤ₅ (the integers modulo 5). This group lives on a clock with 5 hours. Try adding numbers and watch them wrap around! Click two numbers on the clock to add them (mod 5) Complete Group Table (+ mod 5) Symmetry Groups: The Triangle's Secrets This equilateral triangle has hidden symmetries! Apply rotations and reflections to discover D₃ (the dihedral group of order 6). Matrix Groups: Rotation Matrices Rotation matrices form the group SO(2) . Watch how matrices rotate a vector while preserving its length! These seemingly different groups are actually related! Watch as we reveal the hidden connections through group homomorphisms . From clock arithmetic to symmetries to matrix transformations, groups appear in countless contexts. Learning to recognize group structure helps you see mathematical patterns across disciplines. Closure: Combining elements gives another element. Identity: One element leaves others unchanged. Inverses: Every element can be "undone." Associativity: Grouping doesn't matter. Structure Matters More Than Elements

This is the written version of the interactive lesson above. See the full Abstract Algebra course.