1
$\begingroup$

Is there a theream which is a condition on $n\in\mathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?

After spending some time figuring a condition I tried to find some similar thread but didn't find any.

$\endgroup$
  • 1
    $\begingroup$ When $n$ is composite. $\endgroup$ – user10354138 Dec 1 '18 at 13:22
2
$\begingroup$

Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, \forall n \ge3$.

By your question, $D_n \le D_n$. So true for $\forall n\ge3$.

But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$. Hence when $n$ is composite we get a required subgroup.

$\endgroup$
1
$\begingroup$

Any cyclic group has unique subgroup of any order which is divisior of its order

Now in $D_n$ How many subgroup of order 2?

They are at least n .

SO for n>1

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.