Proper condition on the dihedral group

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.

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

2 Answers

Dihedral group $$D_n = , \forall n \ge3$$.

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 $$$$. Hence when $$n$$ is composite we get a required subgroup.

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