For any integer $n \ge 3 $, prove that $D_n$ has a subgroup of order 4 if and only if n is even

Question

Suppose $H \lt D_n$, $|H|=4$

Since $|D_n|=2n$, by Lagrange's Theorem $$4|2n$$ $$2n=4k $$ for some $\in \Bbb Z^+$

$$n=2k$$ thus $n$ is even

Conversely suppose $n$ is even. then how to show $D_n$ has subgroup of order 4 ?

please give me a hint please!

Answer 1

$n$ is even implies $R_{180} \in D_n$ and is of order $2$. So for any reflection $F$, we have $$R_{180}F=FR_{180}$$ and note that any reflection has order $2$ and the latter element has also order $2$.

Hence .................. is your required subgroup!

Answer 2

The dihedral groups $D_n$ are supersolvable, hence the converse of Lagrange's Theorem always holds. Since $4\mid D_{2n}$ for even $n\ge 2$, there exists a subgroup of order $4$.

References:

Dihedral group is supersolvable

Complete classification of the groups for which converse of Lagrange's Theorem holds