So I've found all the cyclic subgroups: $\langle e\rangle$, $\langle r\rangle$, $\langle sr^n\rangle$, and $D_5$ itself (which are 8 subgroups), but how do I know if these are all? How can I find the rest?

  • $\begingroup$ Hint: What are the possible orders of the subgroups? $\endgroup$ – Tobias Kildetoft Feb 10 '15 at 19:24
  • $\begingroup$ @TobiasKildetoft I haven't learnt about Lagrange's theorem yet, so I think I'm supposed to do without it. $\endgroup$ – Frank Vel Feb 10 '15 at 19:25
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    $\begingroup$ Then I guess you just need to show that if you take two elements not in the same cyclic subgroup then the only subgroup containing both is the entire group. $\endgroup$ – Tobias Kildetoft Feb 10 '15 at 19:26

Questions on subgroups of $D_n$ are quite frequent on this site. It is certainly possible to give a short argument each time, but I think that the article of Keith Conrad is worth to be noted, because it gives all the necessary details to solve the questions in general. Section $3$ classifies all subgroups of $D_n$.


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