Statement#

Let $G$ be a cyclic group of order $n$ generated by an element $g$, i.e., $G = \langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}$.

Prove that $G$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$ (the group of integers modulo $n$ under addition).

Required Topics#

• Cyclic groups and generators
• Group homomorphisms and isomorphisms
• First Isomorphism Theorem
• Modular arithmetic

What You Need to Show#

1. Define an explicit isomorphism $\phi: \mathbb{Z}/n\mathbb{Z} \to G$
2. Prove that $\phi$ is a homomorphism
3. Prove that $\phi$ is bijective (one-to-one and onto)

Solution#

Solution coming soon.