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?
Best Answer
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 $\mathbb Z_2\times\mathbb Z_2$ ?