# How do I find all the homomorphisms from $D_4$ to $\Bbb Z_2\times\Bbb Z_2$?

How do I find all the homomorphisms from $D_4$ to $\Bbb Z_2\times\Bbb Z_2$?

This is the rout I have taken so far.

I know that if we are considering a onto homomorphism then there are going to be 4 elements in D4/K where K is the kernel of our homomorphism. So our kernel must have 2 elements. There is only one normal subgroup of order 2 in D4 namely {I,r^2} where R is my rotation. I am now confident that I can construct THE homomorphism from D4 onto Z2 X Z2. I know there will only e one, because we only have one option for our kernel. Now I am confused about how to find the other homomorphisms. There is only one proper subgroup of Z2 x Z2, so that subgroup must be isomorphic to Z2. Now this means our kernel must be 4 elements and normal. All three order 4 subgroups of D4 are normal. So do I then just try and construct the three homomorphisms? and how do I know they will work?

• Does $D_4$ have 4 or 8 elements?
– lhf
Dec 13, 2013 at 0:40
• 8 elements, is the symmetries of a square. Dec 13, 2013 at 0:47
• zzr0ck3r, unfortunately, there are competing notations for dihedral groups: what some call $D_n$ others call $D_{2n}$. Dec 13, 2013 at 0:50
• There are only $4^8\approx 64000$ functions from $D_4$ to $\Bbb Z_2^2$. Just check them all! Dec 13, 2013 at 0:54
• I understand that dfeuer that is why I use D4 because only one thing can be D4. Dec 13, 2013 at 0:57

Hint: Consider this presentation: $D_4=\langle x, y \mid x^4 = y^2 = (xy)^2 = 1 \rangle$.
It is enough to find the images of $x$ and $y$ under a homomorphism, as long as they satisfy the relations above. What are the possibilities when the codomain is $\mathbb Z_2\times\mathbb Z_2$ ?
Hint: The pair $f:A\to B$, $g:A\to C$ determines and is determined by $(f,g):A\to B\times C$. Finally you will have to find all (normal) subgroups of $D_4$ of index $2$.
• @zzr0ck3r You are looking for normal subgroups of $D_4$ with index 2, not order 2. What you need to do is to find all homomorphisms from $D_4$ to $\mathbb{Z}_2$ first (to convince yourself that you have found all of them, just do casework: "if $f: D_4 \to \mathbb{Z}_2$ is a homomorphism, then either this happens, or this happens, etc."), and then consider Berci's suggestion above about what to do when mapping to $\mathbb{Z}_2 \times \mathbb{Z}_2$. Dec 13, 2013 at 0:53