In order to show that the dihedral group $D_n$ is not simple for $n>2$ I am attempting to find a subgroup of $D_n$ that has index $2$ knowing that any subgroup of index 2 is normal. Any hints would be appreciated.
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The dihedral group is the group of symmetries of an $n$-gon. Can you think of a natural $n$-element set (subgroup) of symmetries? – Ethan Bolker Apr 02 '16 at 21:20
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Since $D_n$ is solvable for all $n\ge 1$, see here, it i s also clear that it is not simple. – Dietrich Burde Apr 03 '16 at 08:45
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The cyclic group of order $\;n\;$ of all rotations has always index two in $\;D_{2n}$
DonAntonio
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If $;D_{2n}:= \langle,s,t;:;s^2=t^4=1,,,sts=t^{n-1};\rangle; $ , then $;R:=\langle,t,\rangle;$ – DonAntonio Apr 02 '16 at 21:32