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Group theory and generalizations Definition

Cyclic subgroup generated by an element

Builds on SubgroupLaws of exponents in a group

Definition

Let GG be a group and let g∈Gg \in G. The cyclic subgroup generated by gg is the set ⟨g⟩={gk:k∈ℤ}\langle g \rangle = \{g^k : k \in \mathbb{Z}\} of all integer powers of gg.

Remarks

By the laws of exponents, gj(gk)−1=gj−kg^j (g^k)^{-1} = g^{j-k}, thus ⟨g⟩\langle g \rangle is a subgroup of GG. It is the subgroup generated by the set {g}\{g\}. In additive notation, ⟨g⟩={kg:k∈ℤ}\langle g \rangle = \{kg : k \in \mathbb{Z}\}, for example ⟨3⟩=3ℤ\langle 3 \rangle = 3\mathbb{Z} in (ℤ,+)(\mathbb{Z}, +).

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Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Definition and notation
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Tag0YT586TKXF
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Aliasescyclic-subgroup
Kinddefinition
WritingW0 stub
MSC20A05
DefinesCyclic subgroup generated by an element
UsesSubgroup, Laws of exponents in a group

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