Abstract Algebra/Group Theory/Group/a Cyclic Group of Order n is Isomorphic to Integer Moduluo n with Addition
Theorem
editLet 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
editLet 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 have m elements and f is injective