Exercise 7.2

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Find the finite subgroups of $\R^*$ and $\C^*$ and show directly that they are cyclic.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
If $G$ is a finite subgroup of $\R$ or $\C$, and $n = \vert G \vert$, then from Lagrange's Theorem, $x^n = 1$ for all $x \in G$.

$\bullet$ If  $G$ is a finite subgroup of $\R^*$, then the solutions of $x^n=1$ are in $\{-1,1\}$, so $\{1\} \subset G\subset \{-1,1\}$ : 
$G = \{1\}$ or $G = \{-1,1\}$, both cyclic.

$\bullet$ If  $G$ is a finite subgroup of $\C^*$, then $G \subset \mathbb{U}_n = \{e^{2ik\pi/n}\ \vert \ 0 \leq k \leq n-1\}$. As $\vert G \vert = \vert \mathbb{U}_n \vert = n$, then  $G = \mathbb{U}_n \simeq \Z/n\Z$ is cyclic.
\end{proof}

Exercise 7.2

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Exercise 7.2

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