# Thread: Galois Theory: Prove That The Dihedral Groups Are Solvable

1. ## Galois Theory: Prove That The Dihedral Groups Are Solvable

I seriously haven't got a clue how to even start on this one. Help me?

2. Let $D_{2n} = \langle \sigma, \tau \mid \sigma^4 = e = \tau^2, \tau^{-1} \sigma \tau = \sigma^{-1} \rangle$ be a dihedral group of order $2n$. Then $[D_{2n} : \langle \sigma \rangle] = 2$, so $\langle \sigma \rangle \lhd D_{2n}$. Therefore $\{e\} \lhd \langle \sigma \rangle \lhd D_{2n}$ is a subnormal series of $D_{2n}$ with Abelian factors, so $D_{2n}$ is solvable.

3. Originally Posted by Giraffro
Let $D_{2n} = \langle \sigma, \tau \mid \sigma^4 = e = \tau^2, \tau^{-1} \sigma \tau = \sigma^{-1} \rangle$

Correcting a possible minor typo: it must be $\sigma^n=1\,,\,\,not\,\,\,\sigma^4=1$.

Tonio

be a dihedral group of order $2n$. Then $[D_{2n} : \langle \sigma \rangle] = 2$, so $\langle \sigma \rangle \lhd D_{2n}$. Therefore $\{e\} \lhd \langle \sigma \rangle \lhd D_{2n}$ is a subnormal series of $D_{2n}$ with Abelian factors, so $D_{2n}$ is solvable.
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### dihedral group is solvable

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