How does one find the number of generators of a cyclic group?
I have been given two cyclic groups, one of order 8 and one of order 60. No specifications beyond that. How do I find the number of generators of each?
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The number of generators for a cyclic group of order n is φ(n).
φ(n) denotes Euler’s phi function, which counts the number of positive integers up to n which are relatively prime to n.
Facts:
(i) φ(ab) = φ(a) φ(b) if gcd(a, b) = 1
(ii) φ(p^n) = p^n – p^(n-1) if p is prime.
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1) There are φ(8) = 2^3 – 2^2 = 4 generators for a cyclic group of order 8.
2) There are φ(60) = φ(2^2) φ(3) φ(5) = (2^2 – 2) * 2 * 4 = 16 generators for a cyclic group of order 60.
I hope this helps!
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