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 𝑎 ∈𝐺 we define ⟨𝑎⟩ ={𝑎𝑛 ∣𝑎 ∈𝐺}, where 𝑎0 =𝑒 and 𝑎−𝑛 =(𝑎−1)𝑛.¹ group

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

Properties

• The order of a cyclic group equals the order of its generator, i.e. |𝑎| =|⟨𝑎⟩|.
• 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 ⟨𝑎⟩ =⟨𝑎𝑗⟩ iff |𝑎| and 𝑗 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 ↩