Find all Quotient (or Factor) Groups of $D_4$ (Dihedral Group of order $8$)

Question

I need to be able to find all of the quotient groups for dihedral group 4 with

$D_4 = \{ e,R,R^2,R^3,V,H,D,D'\}$.

I know I have to start by finding the normal subgroups, which are

$\{e,R^2\}$ $\{e,R,R^2,R^3\}$ $\{e,R^2,V,H\}$ $\{e,R^2,D,D'\}$.

Then I need to find the sets defined by $G/H=\{ aH : a \in G\}$ with operation $aH bH = abH$.

I am stuck here; could someone please show me how to find all of the factor groups for D4?

Answer 1

The identity element of the quotient group is the coset which has elements of the normal subgroup. For example $D_4/\{e,R,R^2,R^3\}$ has identity element $\{e,R,R^2,R^3\}$. If the group has order $8$ and the normal subgroup has order $4$, then the order of the quotient group is $2$, so there is only one other element of the quotient group besides the identity -- it's the coset which comes from the group elements not in the normal subgroup.

If the normal subgroup has order $2$, then the quotient subgroup has order $4$. I.e., $D_4/\{e,R^2\}$ has identity $\{e,R^2\}$ and its three cosets $\{R,R^3\}, \{V,H\},$ and $\{D,D'\}.$

It should be noted that you omitted the trivial normal subgroups $\{e\}$ and $D_4.$