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

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## Theorem[edit]

Let C_{m} be a cyclic group of order *m* generated by *g* with

Let be the group of integers modulo m with addition

- C
_{m}is isomorphic to

- C

## Lemma[edit]

Let *n* be the minimal positive integer such that *g*^{n} = *e*

Proof of Lemma

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

1. 2. as *i*-*j*=*sn*+*r*, and*g*^{n}=*e*3. 4. as n is the minimal positive integer such that *g*^{n}=*e*- and 0 ≤
*r*<*n*

5. 0. and 7. 6. - and 0 ≤

## Proof[edit]

- 0. Define

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

- f is homomorphism:

- f is injective by lemma

- f is surjective as both and have
*m*elements and f is injective