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单词 CyclicGroup
释义

cyclic group


A group is said to be cyclic if it is generated by a single element.

Suppose G is a cyclic groupMathworldPlanetmath generated by xG.Then every element of G is equal to xk for some k.If G is infinite, then these xk are all distinct,and G is isomorphicPlanetmathPlanetmathPlanetmath to the group .If G has finite order (http://planetmath.org/OrderGroup) n,then every element of G can be expressed as xk with k{0,,n-1},and G is isomorphic to the quotient groupMathworldPlanetmath /n.

Note that the isomorphisms mentioned in the previous paragraphimply that all cyclic groups of the same order are isomorphic to one another.The infinite cyclic group is sometimes written C,and the finite cyclic group of order n is sometimes written Cn.However, when the cyclic groups are written additively,they are commonly represented by and /n.

While a cyclic group can, by definition, be generated by a single element,there are often a number of different elements that can be used as the generatorPlanetmathPlanetmathPlanetmath: an infinite cyclic group has 2 generators,and a finite cyclic group of order n has ϕ(n) generators,where ϕ is the Euler totient function.

Some basic facts about cyclic groups:

  • Every cyclic group is abelianMathworldPlanetmath.

  • Every subgroupMathworldPlanetmathPlanetmath of a cyclic group is cyclic.

  • Every quotient of a cyclic group is cyclic.

  • Every group of prime order is cyclic. (This follows immediately from Lagrange’s Theorem.)

  • Every finite subgroup of the multiplicative groupMathworldPlanetmath of a field is cyclic.

Titlecyclic group
Canonical nameCyclicGroup
Date of creation2013-03-22 12:23:27
Last modified on2013-03-22 12:23:27
Owneryark (2760)
Last modified byyark (2760)
Numerical id21
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20A05
Related topicGeneralizedCyclicGroup
Related topicPolycyclicGroup
Related topicVirtuallyCyclicGroup
Related topicCyclicRing3
Definescyclic
Definescyclic subgroup
Definesinfinite cyclic
Definesinfinite cyclic group
Definesinfinite cyclic subgroup
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更新时间:2025/5/3 19:42:49