Subgroup

Cyclic subgroup

A cyclic subgroup is the smallest possible subgroup containing some element. A cyclic group may be generated from a single element (the generator) using the inverse and binary operations to “complete” it. Given a generating element we define , where and .¹ group

Every cyclic group is isomorphic to under addition, where is the order of the generator. In the infinite case this is just additive .

Properties

• The order of a cyclic group equals the order of its generator, i.e. .
• All cyclic subgroups are abelian (see above)
• Generators of a finite group From the theorem on Order of powers of a group element, it follows that iff and are coprime.
• Fundamental theorem of cyclic groups
• Number of elements of each order in a cyclic group
• Group of prime order

Bibliography

• 2017, Contemporary Abstract Algebra, §5 (pp. 75ff.)

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Footnotes

1. 2017, Contemporary Abstract Algebra, p. 65 ↩