Abstract Algebra/Group Theory/Group/a Cyclic Group of Order n is Isomorphic to Integer Moduluo n with Addition

Theorem
Let Cm be a cyclic group of order m generated by g with $$\ast$$

Let $$ (\mathbb{Z} / m, +)$$ be the group of integers modulo m with addition


 * Cm is isomorphic to $$ (\mathbb{Z} / m, +)$$

Lemma
Let n be the minimal positive integer such that gn = e


 * $$g^{i} = g^{j} \leftrightarrow i = j~\text{mod}~n $$


 * Let i > j. Let i - j = sn + r where 0 &le; r < n and s,r,n are all integers.

|-    | 1. $$g^{i} = g^{j} \;$$ ||         |-     | ||          |-     | 2. $$ e = g^{i - j} = g^{sn + r} = [g^{n}]^{s} \ast g^{r} = [e]^{s} \ast g^{r} = g^{r}$$ || as i - j = sn + r, and gn = e |-    |3. $$ g^{r} = e $$|| |-    | ||          |-     |4. $$ r = 0 $$ || as n is the minimal positive integer such that gn = e
 * and 0 &le; r < n

|-    | ||          |-     |5. $$ i - j = sn $$ || 0. and 7. |-    |6. $$ i = j~\text{mod}~n $$|| |}

Proof

 * 0. Define  $$\begin{align}

f\colon C_m &\to \mathbb{Z}/m \\ g^{i} &\mapsto i ~\text{mod}~m \end{align}$$


 * Lemma shows f is well defined (only has one output for each input).


 * f is homomorphism:
 * $$f(g^{i}) + f(g^{j}) = i + j ~\text{mod}~m = f(g^{i +j}) = f(g^{i} \ast g^{j})$$


 * f is injective by lemma


 * f is surjective as both $$\mathbb{Z}/m$$ and $$C_{m}$$ have m elements and f is injective