Mathematics.

groups

Cyclic Groups

Abstract Algebra I25 minDifficulty6 out of 10

You should know: subgroups

Overview

A cyclic group is a group that can be generated by repeatedly applying a single element and its inverse. Cyclic groups are the simplest possible groups structurally, and every subgroup of a cyclic group is itself cyclic, making them a natural base case for classification theorems.

Formal Definition

Definition

A group G is cyclic if there exists an element g ∈ G, called a generator, such that every element of G is an integer power of g:

G=⟨g⟩={gn:n∈Z}G = \langle g \rangle = \{ g^n : n \in \mathbb{Z} \}
Cyclic group generated by g
∣G∣=min⁡{n>0:gn=e}|G| = \min\{ n > 0 : g^n = e \}
Order of a finite cyclic group equals the order of its generator

Properties

Abelian

Every cyclic group is abelian: gigj=gi+j=gjgi.\text{Every cyclic group is abelian: } g^i g^j = g^{i+j} = g^j g^i.

Subgroups of cyclic groups are cyclic

If G=⟨g⟩ and H≤G, then H is cyclic.\text{If } G = \langle g \rangle \text{ and } H \leq G, \text{ then } H \text{ is cyclic.}

Classification

Every cyclic group of order n is isomorphic to (Zn,+); every infinite cyclic group is isomorphic to (Z,+).\text{Every cyclic group of order } n \text{ is isomorphic to } (\mathbb{Z}_n, +); \text{ every infinite cyclic group is isomorphic to } (\mathbb{Z}, +).

Generators of ℤₙ

gk generates Zn  ⟺  gcd⁡(k,n)=1, so Zn has φ(n) generatorsg^k \text{ generates } \mathbb{Z}_n \iff \gcd(k, n) = 1, \text{ so } \mathbb{Z}_n \text{ has } \varphi(n) \text{ generators}

Condition: φ = Euler's totient function

Applications

The multiplicative group modulo a prime p is cyclic, and this structure underlies Diffie-Hellman key exchange and ElGamal encryption.

Worked Examples

  1. An element k generates ℤ₆ iff gcd(k, 6) = 1.

    gcd⁡(k,6)=1\gcd(k,6)=1

  2. Check k = 1: gcd(1,6)=1 — generator. k=2: gcd(2,6)=2 — not. k=3: gcd(3,6)=3 — not. k=4: gcd(4,6)=2 — not. k=5: gcd(5,6)=1 — generator.

Answer: The generators of ℤ₆ are 1 and 5, matching φ(6) = 2.

Practice Problems

Difficulty 5/10

Show that every subgroup of (ℤ, +) has the form nℤ for some non-negative integer n.

Summary

  • A cyclic group ⟨g⟩ consists of all integer powers of a single generator g.
  • Every cyclic group is abelian, and every subgroup of a cyclic group is itself cyclic.
  • Finite cyclic groups of order n are all isomorphic to (ℤₙ, +); infinite ones to (ℤ, +).
  • ℤₙ has exactly φ(n) generators, precisely the residues coprime to n.

References

  1. WebsiteWikipedia — Cyclic group