The four homomorphisms created by the direct product construction are each an example of a more general construction of homomorphisms involving groups $G$, $H$ and $G\times H$. By using the same groups as in the example above, see if you can discover and describe these constructions with exact definitions of the four homomorphisms in general.
Your tools for investigating a SageMath group homomorphism are limited, you might take each generator of the domain and see what its image is. Here is an example of the type of computation you might do repeatedly. We’ll investigate the second homomorphism. The domain is the dihedral group, and we will compute the image of the first generator.
G = CyclicPermutationGroup(3) H = DihedralGroup(4) results = G.direct_product(H) phi = results[2] print( H. gens() ) a = H.gen(0) print( a ) print( phi(a) )
Sage Exercises. Stephen F. Austin State University, 29 Sept. 2021, https://math.libretexts.org/@go/page/81122.
I don't know how to approach the above to make clear definitions of the returned homomorphisms. I am aware of the 5 constructions returned: The first being the direction product of $G$ and $H$, the remaining 4 being homomorphism from $G \rightarrow G\times H, H \rightarrow G \times H, G\times H \rightarrow G$ and $G\times H \rightarrow H $. I used the orders of the images, kernels, domains and codomains, to determine the first two are injective and the remaining are surjective. I don't know how to approach the generator angle. Documentation concerning the direct_product function is helpful either.
From my understanding, the homomorphism definitions can be made exact by determining the images of the generators of the domain for a particular group domain under the mapping?
I think the question is asking this: given groups $G$ and $H$, there are four standard homomorphisms involving $G$, $H$, and $G \times H$. Can you describe them? For example, there is supposed to be a homomorphism $\phi: H \to G \times H$. If you want to describe it, you should specify $\phi(h)$ for any element $h \in H$. $\phi(h)$ must be an element of $G \times H$; which element? Similarly, to describe $\psi: G \times H \to G$, you need to specify $\psi(g,h)$ (which should be an element of $G$) for any $(g,h) \in G \times H$.
If you specify where the generators of a group go under a homomorphism, that is enough to specify what a homomorphism does in general, and so for the particular groups in the example, it is certainly good enough. I think the question is asking whether you can figure out what happens for any groups $G$ and $H$, not just these two particular ones.