Clock Groups

Generators and star polygons

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Clock arithmetic forms the cyclic group ℤₙ.
• A step k generates ℤₙ exactly when gcd(k, n) = 1; otherwise it makes a smaller subgroup.
• Every cyclic group is a clock in disguise; ℤₙ has φ(n) generators.

Steps

1. Clock arithmetic
2. Step around the clock
3. The order of an element
4. Prime clocks
5. Every cyclic group is a clock
6. How many generators?