Subgroups of $ \mathbb{R}^{} $ and $ \mathbb{C}^{} $ under multiplication

Question

I was studying subgroups then I asked myself what are the subgroups of $ \mathbb{R}^{*} $ and $ \mathbb{C}^{*} $ under multiplication

In my search i've found that $\alpha\mathbb{R}^{*} $ where $\alpha$ is a real number different than zero but it was unconvincing and for $ \mathbb{C}^{*} $ I've found only topics that treated finite subgroups.

Thanks, for reading my question.

Under multiplication or addition of $\mathbb{R}$ and $\mathbb{C}$? — Qurultay, Mar 26 '18 at 18:27
Then you may consider $\mathbb{R}^\times=\mathbb{R}-{0}$ and $\mathbb{C}^\times=\mathbb{C}-{0}$. Because $0$ is not invertible under multiplication — Qurultay, Mar 26 '18 at 18:34
Sorry i missed that, thanks i will edit the question. Thanks — ThePirateKing, Mar 26 '18 at 18:36
Well, for $\mathbb{R}^\times$, we see, for any $a\in\mathbb{R}^\times$, the cyclic subgroups ${a^n:n\in\mathbb{Z}}$, as well $\mathbb{Q}^\times$ — Qurultay, Mar 26 '18 at 18:45

score 2 · Accepted Answer · answered Mar 26 '18 at 18:49

2

Actually, since $\exp:\mathbb{R}\rightarrow \mathbb{R}^*$ is an injective group homomorphism, $(\mathbb{R},+)$ is a subgroup of $\mathbb{R}^*$, and all subgroups of this as well. And the subgroups of $(\mathbb{R},+)$ are up to isomorphism the torsion-free abelian groups of rank $\alpha$ for every cardinal $\alpha\leq 2^{\aleph_0}$.

answered Mar 26 '18 at 18:49

Dietrich Burde

130,978

and for $\mathbb{C} $ ? – ThePirateKing Mar 26 '18 at 19:16
Have look at this site, e.g., here. Note that $\mathbb{C}^*=\mathbb{R}\times \mathbb{R}/\mathbb{Z}$. – Dietrich Burde Mar 26 '18 at 19:24

Subgroups of $ \mathbb{R}^{*} $ and $ \mathbb{C}^{*} $ under multiplication

1 Answers1

Subgroups of $ \mathbb{R}^{} $ and $ \mathbb{C}^{} $ under multiplication