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I saw this question earlier on the forum and was wondering if my result to it was correct!

If D14 is the dihedral group acting on a heptagon, are the only subgroups in the lattice D14, < r> , < o> and e? Where o is rotation and r is reflection and < x> gives the group generated by x.

My thought process was that < o^n> would not be unique as it would generate exactly the some thing as < o> and everything else would generate the entire group so not be a subgroup. Is my thinking correct?

Am I also right in thinking the centre is just (e) and the normal subgroups are just D14, (e) and < o>.

Thanks =)

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  • $\begingroup$ If you meant $r$ to run through all the reflections, then you've got it! If $r$ represents just one of the reflections, then you're missing some subgroups. $\endgroup$
    – Ben Blum-Smith
    Commented Dec 4, 2014 at 21:14
  • $\begingroup$ I had r as reflection in a line from a single corner to the opposite side. Wouldn't r^2 just give back the identity element so there is only one reflection? $\endgroup$
    – user2973447
    Commented Dec 4, 2014 at 21:29
  • $\begingroup$ There are $n$ reflections. $r$, $or$, $o^2r$, etc. $\endgroup$
    – Ben Blum-Smith
    Commented Dec 8, 2014 at 21:50
  • $\begingroup$ Note that the heptagon has 7 corners so there are 7 reflections in the lines that you describe. $\endgroup$
    – Ben Blum-Smith
    Commented Dec 8, 2014 at 21:51

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You're right with the centre being {e} as this is true for all groups $D_n$ where n is odd.

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