In our lecture we were asked to answer the following questions:
How many are there homomorphisms from $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ to $S_{3}$? And vice-versa?
How many are there homomorphisms from $S_{3}$ to $Q_{8}$? And vice-versa?
I am a little confused about it. I know from the definition of a homomorphism that it is a map from one group to another such that the group operation is preserved. But how am I supposed to find one between two groups?
I know the group $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ is $(0,0), (1,1), (1,0), (0,1)$ and the symmetric group $(1), (12), (13), (132), (123)$ but how do I know which elements to map to which elements?
Am I supposed to look at underlying properties of the groups? Such as whether it is abelian or not, or whether or not it is a cyclic group? And then compare generators? Could someone explain this concept to me using the above question?
Hint 1: homomorphisms are determined by where they send generators.
Explanation: Let $\phi:X \to Y$ be a homomorphism, where $X = \langle x_1,\ldots, x_n \rangle$. The above hint means that if another homomorphism $\psi$ sends all the $x_i$ to the same place in $Y$, then $\phi \equiv \psi$ identically. To see this, take any arbitrary element $x \in X$. Then $x =\prod x_i$ is some finite product of generators. So $$\phi(x) = \phi \left( \prod x_i \right) = \prod \phi(x_i) = \prod \psi(x_i) = \psi\left(\prod x_i\right) = \psi(x)$$ That is, $\phi$ and $\psi$ act identically on $x$. Said another way, they are the same. And yet another way: homomorphisms are determined by how they act on generators. This + the hint from @Dave on preserving order should take you far since the example groups are small.
Hint 2: using the first isomoprhism theorem will help count as well since you know a homomorphism gives you a kernel, and that kernel is a subgroup of your domain. How many subgroups are there? Are each a possible kernel?