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If $S$ is the cyclic subgroup of order $n$ in the dihedral group $D_n$, show that $D_n/S$ is isomorphic to $Z_2$.

I know I'm supposed to find an epimorphism from $D_n$ to $Z_2$, such that $S$ is its Kernel, so that the quotient group $D_n/S$ will be isomorphic to $Z_2$. But I have no idea of how to find such an isomorphism. Besides, I don't even know what $n$ is, so I have no way of finding the elements of $D_n$ and defining an epimorphism on them.

Any help will be appreciated!!!!

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2 Answers 2

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$D_n$ has order $2n$. If you have a subgroup $S$ of order $n$, then $D_n/S$ will have order 2 and hence is isomorphic to $\mathbb{Z}_2$.

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  • $\begingroup$ Thanks!!!I haven't thought of that... I've been thinking about finding an isomorphism $\endgroup$ Nov 3, 2011 at 2:14
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    $\begingroup$ It should perhaps be noted that subgroups of index 2 are always normal, so the quotient group exists. $\endgroup$ Nov 3, 2011 at 2:41
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I'm not sure what you mean when you say you have no way of finding the elements of $D_n$. What definition have you been given for this group?

$D_n$ can be defined as the group of symmetries of a regular $n$-sided polygon. It has $2n$ elements; $n$ rotations, and $n$ "flips", or reflections. The rotations form a subgroup; that's your $S$. If you can prove that it's a normal subgroup, then you know the order of the quotient group, and you know there is, up to isomorphism, only one group of that order, and you're done.

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  • $\begingroup$ I mean, since I don't know what n is, I don't know how to write Dn explicitly as {1,R,R2,...,Rn-1, D1,...,Dn} if n is odd there are n reflections around each vertex, if n is even there are n/2 such reflections and n/2 mirror reflections around an edge so it's different when n changes $\endgroup$ Nov 3, 2011 at 1:59
  • $\begingroup$ @Scharfschütze: For the sake of concreteness, take the complex nth roots of unity as the corners of the regular n-gon. Rotations in the complex plane then correspond to multiplying by one of these roots (multiplying by 1 being of course the identity of the transformations). All the other dihedral group elements are reflections, as Gerry says. You can represent them as complex conjugation followed by a rotation if you like. $\endgroup$
    – hardmath
    Nov 3, 2011 at 15:03

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