> [!NOTE] **Definition** (Cyclic Group) > The [[Groups|group]] $G$ is cyclic iff it is the [[Generated Subgroup|subgroup generated]] by some element $g\in G,$ denoted $G=\langle g \rangle$ We say that $g$ is the *generator* of $G.$ # Properties Firstly, the [[Order of Group Element Equals Order of Subgroup Generated by Element|order of a cyclic group is the same as the order of its generator]]. Also note that [[Cyclic Groups are Abelian|cyclic groups are abelian]]. The [[Fundamental Theorem of Cyclic Groups|fundamental theorem of cyclic groups]] asserts that every *subgroup* of a cyclic group is also cyclic. # Applications **Examples**: The [[Roots of Unity|roots of unity]] are a multiplicative cyclic group. The set of [[Integers|integers]] with addition form an *infinite* cyclic group generated by $1$ or $-1.$ The [[Integers modulo n|integers modulo n]] with modular addition form a *finite* cyclic group generated by $[1]_{n}$. **Cryptography**: [[Discrete Logarithm Problem (DLP)]]. [[Diffie-Hellman Problem]].