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

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Theorem

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Let Cm be a cyclic group of order m generated by g with  

Let   be the group of integers modulo m with addition

Cm is isomorphic to  

Lemma

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Let n be the minimal positive integer such that gn = 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 gn = e
3.  

4.   as n is the minimal positive integer such that gn = e
and 0 ≤ r < n

5.   0. and 7.
6.  

Proof

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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