I need to determine the subgroups of the dihedral group of order 8, $D_4$.

Question

I need to determine the subgroups of the dihedral group of order 4, $D_4$.

I know that the elements of $D_4$ are $\{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$

But I don't understand how to get the subgroups..

Answer 1

By Lagrange's Theorem, the possible orders are $1, 2, 4,$ and $8$.

The only subgroup of order $1$ is $\{1\}$ and the only subgroup of order $8$ is $D_4$.

If $D_4$ has an order $2$ subgroup, it must be isomorphic to $\mathbb{Z}_2$ (this is the only group of order $2$ up to isomorphism). Such a group is cyclic, it is generated by an element of order $2$. Are there any such elements in $D_4$?

If $D_4$ has an order $4$ subgroup, it must be isomorphic to either $\mathbb{Z}_4$ or $\mathbb{Z}_2\times\mathbb{Z}_2$ (these are the only groups of order $4$ up to isomorphism). In the former case, the group is cyclic, it is generated by an element of order $4$. Are there any such elements in $D_4$? In the latter case, the group is generated by two commuting elements of order $2$. Are there any such pairs of elements in $D_4$?

In summary, first find all the elements of order $2$ and all the elements of order $4$; each of them generates a cyclic subgroup. Then consider pairs of elements of order $2$ to find which of them generate subgroups isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$.