It is not too hard to see that for general groups $K(G\times H,d)\cong K(G,d)\times K(H,d)$. So three follows from the first two with the product CW structure.

Edit:

Computing the cohomology of the $K(G,n)$'s on the nose is difficult I think. I think I read somewhere that the full computation was done by Serre.

To get a grasp on the cohomology you look at the pathspace fibration:

$\Omega(K(g,n)\rightarrow PK(g,n)\rightarrow K(g,n)$

and the associated Leray-Serre spectral sequence. As the loopspace of $K(G,n)$ is a model of $K(G,n-1)$ and the pathspace is contractible one gets an inductive description of the cohomology.

It is not too hard to see that for general groups $K(G\times H,d)\cong K(G,d)\times K(H,d)$. So three follows from the first two with the product CW structure.

Edit:

Computing the cohomology of the $K(G,n)$'s on the nose is difficult I think. 

I think I read somewhere that the full computation of the cohomology was done by Serre (edit Cartan apparently).

To get a grasp on the cohomology you look at the pathspace fibration:

$\Omega(K(g,n))\rightarrow PK(g,n)\rightarrow K(g,n)$

and the associated Leray-Serre spectral sequence. As the loopspace of $K(G,n)$ is a model of $K(G,n-1)$ and the pathspace is contractible one gets an inductive description of the cohomology.

It is not too hard to see that for general groups $K(G\times H,d)\cong K(G,d)\times K(H,d)$. So three follows from the first two with the product CW structure.

It is not too hard to see that for general groups $K(G\times H,d)\cong K(G,d)\times K(H,d)$. So three follows from the first two with the product CW structure.

Edit:

Computing the cohomology of the $K(G,n)$'s on the nose is difficult I think. I think I read somewhere that the full computation was done by Serre.

To get a grasp on the cohomology you look at the pathspace fibration:

$\Omega(K(g,n)\rightarrow PK(g,n)\rightarrow K(g,n)$

and the associated Leray-Serre spectral sequence. As the loopspace of $K(G,n)$ is a model of $K(G,n-1)$ and the pathspace is contractible one gets an inductive description of the cohomology.