How would I find the number of distinct homomorphisms and isomorphisms mapping the Klein four group to the Klein four group? How would I find the number of distinct homomorphisms and isomorphisms from the Klein four group to the Klein four group?
Thank you 

Note that $K_4=\mathbb{Z}_2\times \mathbb{Z}_2$  (thinking of $\mathbb{Z}_2$ as $\{0,1\}$ with addition) is generated by the two elements $(0,1)$ and $(1,0)$. Now let $f: K_4\to K_4$ be a homomorphism. If I determine what $f(0,1)$ and $f(1,0)$ is the map $f$ is fully known. Now $f(0,1)$ has four choices and so does $f(1,0)$. So there are $16$ homomorphisms.
For isomorphisms you need $f$ to be injective. So $f(1,0), f(0,1)\neq (0,0)$ and also $(0,0)\neq f(1,1)=f(1,0)+f(0,1)$. Let $f(1,0)=(a,b)$ and $f(0,1)=(c,d)$. Then the last condition means either $a+c\neq 0$ or $b+d\neq 0$. The possibilities for both being non-zero (with $(a,b)\neq (0,0)$ and $(c,d)\neq (0,0)$) are


*

*$a=1$, $b=0$, $\Longrightarrow c=0$, $d=1$. (This gives the identity map)

*$a=0$, $b=1$, $\Longrightarrow c=1$, $d=0$. (This gives the swap map)


The possibilities for $a+c=0$ but $b+d\neq 0$ are


*

*$a=1$, $b=0$, $\Longrightarrow c=1$, $d=1$.

*$a=0$, $b=1$, $\Longrightarrow c=0$, $d=0$ (Impossible).

*$a=1$, $b=1$, $\Longrightarrow c=1$, $d=0$.


Similarly for $a+c\neq 0$ but $b+d= 0$


*

*$a=1$, $b=0$, $\Longrightarrow c=0$, $d=0$ (Impossible).

*$a=0$, $b=1$, $\Longrightarrow c=1$, $d=1$ 

*$a=1$, $b=1$, $\Longrightarrow c=0$, $d=1$.


So we have found a total of 6 candidates for isomorphisms. It is easy to check that all indeed give an isomorphism (need to check surjectivity).
