Cyclic subgroup

From Maths
Jump to: navigation, search


A cyclic subgroup is a generated subgroup, where the generating set is a single element of the group [ilmath](G,\times)[/ilmath], that is:

  • For any [ilmath]g\in G[/ilmath] the cyclic subgroup generated by [ilmath]g[/ilmath] is [ilmath]\langle g\rangle[/ilmath] (for the meaning of this notation see Generated subgroup)
  • This is equivalent to [math]\langle g\rangle=\{g^n\ |\ n\in\mathbb{Z}\}[/math]

This is a subgroup, and is Abelian (for finite groups - not sure about infinite)

TODO: Proof of claims