Knowlpedia

Cyclic Subgroup

The subgroup consisting of all integer powers of one group element.

Let GG be a and g∈Gg\in G. The cyclic subgroup generated by gg is

⟨g⟩={gn:n∈Z}.\langle g\rangle=\{g^n:n\in\mathbb Z\}.

It is the smallest of GG containing gg, hence a special case of the .

A group is cyclic if it equals ⟨g⟩\langle g\rangle for some gg.

Examples
  • In (Z,+)(\mathbb Z,+), ⟨3⟩=3Z\langle 3\rangle=3\mathbb Z.
  • In C×\mathbb C^\times, ⟨i⟩={1,i,−1,−i}\langle i\rangle=\{1,i,-1,-i\}.
  • The subgroup ⟨e⟩={e}\langle e\rangle=\{e\} is trivial.