Dense cyclic subgroup - MathOverflow

Dense cyclic subgroup

2010-08-31T09:24:27Z

Does anyone know a continuous group (not necessarily locally compact) with dense cyclic subgroup other than a torus?

Answer by Charles Matthews for Dense cyclic subgroup

2010-08-31T09:56:02Z

How about the Bohr compactification of the infinite cyclic group?

Answer by Xandi Tuni for Dense cyclic subgroup

2010-08-31T09:58:07Z

How about the infinite cyclic group itself with the discrete topology? Or p-adic integers?

Answer by Gjergji Zaimi for Dense cyclic subgroup

2010-08-31T10:10:02Z

You already have some examples in the other answers. Groups which have a dense cyclic subgroup are called Monothetic groups. In the article "On monothetic groups" by P.R. Halmos and H. Samelson, you can find many of their properties, such as

Every compact connected separable (abelian) group is monothetic.

Answer by Keivan Karai for Dense cyclic subgroup

2010-08-31T10:29:40Z

First, it is clear the group has to be abelian. Now, if you assume that $G$ is locally compact, then by the classification you can decompose $G$ as $G={\mathbb R}^n \times H$ where $H$ has a compact open subgroup. Clearly, there can be no ${\mathbb R}^n$ factor, so $G$ has a compact open subgroup. Now, suppose $G$ is itself compact and topologically generated by $g$. Then any character $\chi$ in the dual of $G$ vanishing on $g$ will be identically zero. So, the map $\chi \mapsto \chi (g)$ is injective, hence the dual is a subgroup of $U(1)$. Conversely, you can also see that if $\Gamma$ is a subgroup of $U(1)$ (considered with the discrete topology) then the dual of $\Gamma$ has a dense cyclic subgroup. By taking various subgroups you can, for instance, get the $p$-adic integers, or the n-torus.