How to find number of abelian subgroups of diheral group?

Question

How to find number of abelian subgroups of diheral group $D_n $?

Attempt: I have counter-examples for $n=1,2$ so I know that it isn't true for $n<3$. Is it true for $n\ge 3$? How do you know this?

Answer 1

Hint 1: any element $a$ of order $2$ generates the subgroup $\{e,a\}$ where $e$ is the identity. A subgroup of order 2 is obviously abelian.

Hint 2: $D_n$ contains a subgroup that is isomorphic to the cyclic group $\mathbb{Z}_n$. This is also abelian.

Answer 2

In addition to the other answers I have three further hints:

Hint 3: by Lagrange, the order of a subgroup divides $2n$.

Hint 4: For $n$ odd, all abelian subgroups are cyclic. For $n$ even, every abelian subgroup is either cyclic or isomorphic to $C_2\times C_2$.

Hint 5: This site offers even a classification of all possible subgroups of $D_n$. Keith Conrad has written a nice note on this topic.