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Thm. 4.2 says that Let a be an element of order n in a group and let k be a positive integer. Then <math><a^k> = <a^{gcd(n,k)}></math> and <math> |a^k| = n/gcd(n,k)</math>. | Thm. 4.2 says that Let a be an element of order n in a group and let k be a positive integer. Then <math><a^k> = <a^{gcd(n,k)}></math> and <math> |a^k| = n/gcd(n,k)</math>. | ||
| − | Let G = <a> be a cyclic group of order n. Then G = <math><a^k></math> if and only if gcd(n,k)=1 | + | Corollary 2 says Let G = <a> be a cyclic group of order n. Then G = <math><a^k></math> if and only if gcd(n,k)=1 |
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| + | Corollary 3 says an integer k in <math>Z_n</math> is a generator of <math>Z_n</math> if and only if gcd(n,k) =1 | ||
Revision as of 15:07, 20 September 2008
The back of the book gives an answer, but I don't find it helpful. Does anyone have a good explaination on how to work this problem?
--Akcooper 16:34, 17 September 2008 (UTC)
Thm. 4.2 says that Let a be an element of order n in a group and let k be a positive integer. Then $ <a^k> = <a^{gcd(n,k)}> $ and $ |a^k| = n/gcd(n,k) $.
Corollary 2 says Let G = <a> be a cyclic group of order n. Then G = $ <a^k> $ if and only if gcd(n,k)=1
Corollary 3 says an integer k in $ Z_n $ is a generator of $ Z_n $ if and only if gcd(n,k) =1
