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Math Help - Galois Theory: Prove That The Dihedral Groups Are Solvable

  1. #1
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    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?
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  2. #2
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    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.
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  3. #3
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    Quote Originally Posted by Giraffro View Post
    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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