Cyclic Groups

Principle

A cyclic group is a group whose elements are all produced by repeatedly applying the group operation to one element. That element is called a generator.

Cyclic groups model repeated processes, such as applying the same rotation again and again until a physical configuration returns to its start.

Notation

• \(g\) is a generator candidate.
• \(g^n\) is the result of applying the group operation repeatedly using \(g\).
• \(g^0=e\), where \(e\) is the identity element.
• \(\langle g\rangle\) is the set generated by \(g\).
• The order of an element is the smallest positive integer \(n\) such that \(g^n=e\), if such an integer exists.
• \(r\) is a rotation used as a generator in examples.

Method

To test whether a finite group is cyclic:

1. Choose an element \(g\) as a generator candidate.
2. List the powers \(g^0,g^1,g^2,\\ldots\).
3. Stop when the identity \(e\) first repeats.
4. Compare the listed elements with the whole group.
5. If every group element appears, then \(g\) generates the group.

Rules

• In multiplicative notation, \(g^m g^n=g^{m+n}\).
• A finite cyclic group repeats after the generator reaches the identity again.
• The repeated identity at the end of the cycle is not counted as a new element.
• Every cyclic group is abelian, so its elements commute.

Examples

Let \(r\) be a \(120^\circ\) rotation of an equilateral triangle. Repeated rotations by \(120^\circ\) produce all rotational symmetries of the triangle.

The first return to the identity occurs at \(r^3=e\). The distinct elements before that repeat are \(e\), \(r\), and \(r^2\).

Therefore the rotational symmetry group of the equilateral triangle is cyclic of order \(3\), generated by \(r\).

Checks

• Stop at the first return to the identity.
• Do not count the repeated identity twice.
• Check that the powers of the generator produce every element, not just several elements.
• Distinguish the order of one element from the number of elements in a larger group.