Exercise 4.4.9 (Every subgroup of $\langle r \rangle$ is normal in $D_{2n}$)

Question

If $r,s$ are the usual generators for the dihedral group $D_{2n}$, use the preceding two exercises to deduce that every subgroup of $\langle r \rangle$ is normal in $D_{2n}$.

Richard Ganaye · Accepted Answer

\begin{proof} 
$K = \langle r \rangle$ is a cyclic subgroup, thus for all divisor $d$ of $n = |H|$, there is exactly one group of order $d$. By exercise $7$, every subgroup $H \leq K$ is characteristic in $K$. Moreover, $K$ has order $n = |D_{2n}|/2$, therefore $K \unlhd G = D_{2n}$. By Exercise 8 part (a), $H \unlhd G$.

Every subgroup of $\langle r \rangle$ is normal in $D_{2n}$.
\end{proof}

Exercise 4.4.9 (Every subgroup of $\langle r \rangle$ is normal in $D_{2n}$)

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Exercise 4.4.9 (Every subgroup of $\langle r \rangle$ is normal in $D_{2n}$)

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