411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Group/a Cyclic Group of Order n is Isomorphic to Integer Moduluo n with Addition
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Theorem
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
Let n be the minimal positive integer such that gn = e
right
Proof of Lemma
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Let i > j. Let i - j = sn + r where 0 ≤ r < n and s,r,n are all integers.
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| 1.
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| 2.
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|3. ||
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|4.
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and 0 ≤ r < n
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|5. || 0. and 7.
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Proof
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