# When are $\langle r,R^2\rangle$ and $\langle rR,R^2\rangle$ proper subgroups of $D_n$?

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?

• 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. Apr 8, 2013 at 13:13
• 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). Apr 8, 2013 at 13:19
• 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. Apr 8, 2013 at 13:23
• 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. Apr 8, 2013 at 13:28
• I wish you had used the following presentation instead: :-) $$D_n=D_{2n}=\langle x,y\mid x^n=y^2=(xy)^2=1\rangle$$ Apr 8, 2013 at 14:03

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 $D_{n/2}$.