# Abstract Algebra/Group Theory/Cyclic groups/Definition of a Cyclic Group

- A cyclic group generated by g is

- where

- Induction shows:

## A cyclic group of order n is isomorphic to the integers modulo n with addition

edit## Theorem

editLet 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

editLet *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