Abstract Algebra/Group Theory/Cyclic groups/Definition of a Cyclic Group
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 A cyclic group generated by g is


 where



 Induction shows:

a Cyclic Group of Order n is Isomorphic to the Integers Moduluo n with Addition[edit]
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
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.
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 has m elements and m is injective