Finite Cyclic Group is Isomorphic to Integers under Modulo Addition

Theorem
Let $\struct {G, \circ}$ be a finite group whose identity element is $e$.

Then $\struct {G, \circ}$ is cyclic of order $n$ $\struct {G, \circ}$ is isomorphic with the additive group of integers modulo $n$ $\struct {\Z_n, +_n}$.

Necessary Condition
Let $\struct {G, \circ}$ be a cyclic group of order $n$.

From List of Elements in Finite Cyclic Group:
 * $G = \set {a^0, a^1, a^2, \ldots, a^n}$

where $a^0 = e, a^1 = a$.

From the definition of integers modulo $n$, $\Z_n$ can be expressed as:


 * $\Z_n = \set {\eqclass 0 n, \eqclass 1 n, \ldots, \eqclass {n - 1} n}$

where $\eqclass x n$ is the residue class of $x$ modulo $n$.

Let $\phi: G \to \Z_n$ be the mapping defined as:
 * $\forall k \in \set {0, 1, \ldots, n - 1}: \map \phi {a^k} = \eqclass k n$

By its definition it is clear that $\phi$ is a bijection.

Also:

Thus the morphism property of $\phi$ is demonstrated, and $\phi$ is thus a group homomorphism.

By definition, a group isomorphism is a group homomorphism which is also a bijection.

Sufficient Condition
Now suppose $G$ is a group such that $\phi: \Z_n \to G$ is a group isomorphism.

Let $a = \map \phi {\eqclass 1 n}$.

Let $g \in G$.

Then $g = \map \phi {\eqclass k n}$ for some $\eqclass k n \in \Z_n$.

Therefore:

So every element of $G$ is a power of $a$.

So by definition $G$ is cyclic.