cyclic group
A group is said to be cyclic if it is generated by a single element.
Suppose G is a cyclic group generated by x∈G.
Then every element of G is equal to xk for some k∈ℤ.
If G is infinite, then these xk are all distinct,
and G is isomorphic
to the group ℤ.
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 group
ℤ/nℤ.
Note that the isomorphisms mentioned in the previous paragraph imply 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 ℤ and ℤ/nℤ.
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 generator: 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:
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•
Every cyclic group is abelian
.
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•
Every subgroup
of a cyclic group is cyclic.
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•
Every quotient of a cyclic group is cyclic.
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•
Every group of prime order is cyclic. (This follows immediately from Lagrange’s Theorem.)
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•
Every finite subgroup of the multiplicative group
of a field is cyclic.
Title | cyclic group |
Canonical name | CyclicGroup |
Date of creation | 2013-03-22 12:23:27 |
Last modified on | 2013-03-22 12:23:27 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 21 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20A05 |
Related topic | GeneralizedCyclicGroup |
Related topic | PolycyclicGroup |
Related topic | VirtuallyCyclicGroup |
Related topic | CyclicRing3 |
Defines | cyclic |
Defines | cyclic subgroup |
Defines | infinite cyclic |
Defines | infinite cyclic group |
Defines | infinite cyclic subgroup |