How do you show that {$ 1,a,a^{2}, ..., a^{n-1} $} is a normal subgroup of the nth dihedral group? I thought maybe it is the only subgroup of n elements , so then it's a normal subgroup. But I don't know how to prove it.
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3Any subgroup of index 2 is a normal subgroup (suppose not, see what happens). – Chappers Oct 07 '15 at 17:24
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Alternatively, you can prove it directly: the dihedral group is generated by $a$ together with a reflection $r$, so all you need to do is to show that $ra^kr^{-1}\in \langle a\rangle$. – rogerl Oct 07 '15 at 17:42