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Here is the question:

When are the groups $\langle r,R^2\rangle$ and $\langle rR,R^2\rangle$ proper subgroups of $D_n$? Your answers should depend on $n$.

When $\langle r,R^2\rangle$ and $\langle rR,R^2\rangle$ are proper subgroups of $D_n$, what are their indices in $D_n$? Prove your answer.

I know how to show a group is subgroup of another group, but what is proper subgroup? and how to show that?

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  • $\begingroup$ A LaTeX tip: < and > mean "less than" and "greater than", and produce spacing correct for that meaning only. When you want angle brackets, you need to use \langle and \rangle. $\endgroup$ Apr 8, 2013 at 13:13
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    $\begingroup$ What are $r$ and $R$? A proper subgroup is a subgroup which does not equal the whole group (you could have consulted any book on group theory, or google). $\endgroup$ Apr 8, 2013 at 13:19
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    $\begingroup$ Seconding @Martin, you must explain your notation if you hope to get any help here. Even the notation $D_n$ is not completely standard; some texts use it for the dihedral group with $2n$ elements, and some use it for the dihedral group with $n$ elements, $n$ being assumed even. $\endgroup$ Apr 8, 2013 at 13:23
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    $\begingroup$ sorry for being unclear, $r$ is the basic reflection in $D_n$, and $R$ is the basic rotation in $D_n$, $D_n$ is dihedral group of an object with n-gon, n can be odd or even, and the order of $D_n$ is 2n. @Zev Chronoles: thank for helping me with the editing, I'm new to this website, so there are alot of thing that I don't know. $\endgroup$ Apr 8, 2013 at 13:28
  • $\begingroup$ I wish you had used the following presentation instead: :-) $$D_n=D_{2n}=\langle x,y\mid x^n=y^2=(xy)^2=1\rangle$$ $\endgroup$
    – Mikasa
    Apr 8, 2013 at 14:03

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If $n$ is odd, then $R^2$ generates all the rotations, so $r$ with $R^2$ generates the whole group. So does $rR$ with $R^2$; in fact, I don't see where there's going to be any difference between $r$ and $rR$, as they are both reflections, and I don't see what makes any one reflection more "basic" than another.

If $n$ is even, $R^2$ generates only half the rotations, and together with $r$, or with $rR$, it generates half of $D_n$, a half isomorphic to $D_{n/2}$.

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