Difference between revisions of "Cyclic group"
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BRichtigen (talk | contribs) (New page: A group ''G'' is '''cyclic''' if it is generated by a single element <math>a \in G</math>, i.e., <math>G = \{a^n, n \in \mathbb{Z} \}</math>. Every cyclic group is...) |
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A [[Group (mathematics)|group]] ''G'' is '''cyclic''' if it is generated by a single element <math>a \in G</math>, i.e., <math>G = \{a^n, n \in \mathbb{Z} \}</math>. | A [[Group (mathematics)|group]] ''G'' is '''cyclic''' if it is generated by a single element <math>a \in G</math>, i.e., <math>G = \{a^n, n \in \mathbb{Z} \}</math>. | ||
| − | Every cyclic group is [[commutative]]. Any group with a [[prime number]] of elements is cyclic. A cyclic group is either [[Isomorphism|isomorphic]] to <math>\mathbb{Z}</math> , or | + | Every cyclic group is [[commutative]]. Any group with a [[prime number]] of elements is cyclic. A cyclic group is either [[Isomorphism|isomorphic]] to <math>\mathbb{Z}</math> , or to <math>\mathbb{Z}/n\mathbb{Z}</math> for some <math>n \in \mathbb{N}</math>. |
Revision as of 23:11, November 17, 2008
A group G is cyclic if it is generated by a single element , i.e., .
Every cyclic group is commutative. Any group with a prime number of elements is cyclic. A cyclic group is either isomorphic to , or to for some .
