Cyclic Group

definitiongroup-theory

Definition

Say a group $(G,\cdot )$ is cyclic generated by $x$ (or $(A,+)$ is cyclic generated by $x$) if $G=\{x^n:n\in \mathbb{Z}\}$ (respectively $A=\{nx:n\in \mathbb{Z}\}$), allowing repetitions. Denote $(G,\cdot )$ or $(A,+)$ as $⟨x⟩$.

Examples

1. The trivial cyclic group generated by $e$.
2. $({\mathbb{Z}}_6,+)$ ${\mathbb{Z}}_6=\{0,1,2,3,4,5\}$ $⟨0⟩=\{0\}$ $⟨1⟩=\{0,1,2,3,4,5,0,\dots \}=\{0,1,2,3,4,5\}={\mathbb{Z}}_6$ $⟨2⟩=\{0,2,4,0,2,\dots \}=\{0,2,4\}\ne {\mathbb{Z}}_6$ $⟨3⟩=\{0,3,0,3,\dots \}=\{0,3\}\ne {\mathbb{Z}}_6$ $⟨4⟩=\{0,4,-2,2,0,\dots \}=\{0,2,4\}\ne {\mathbb{Z}}_6$ $⟨5⟩=\{0,5,4,3,2,1,0,\dots \}=\{0,1,2,3,4,5\}={\mathbb{Z}}_6$ So ${\mathbb{Z}}_6$ is cyclic with generator $1$ or $5$.

Theorems