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 modulo n with addition[edit]

Theorem[edit]

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[edit]

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[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