Group homomorphisms are fundamental. However, I think many undergraduates including myself will benefit from several informal observations about them.
In this post I am not interested in endomorphisms and automorphism.
My feelings are the following:
Isomorphisms are important because the allow to use additional (if it exists of course) structure (e.g. geometric) on an isomorphic group. As long as an isomorphic is established we can learn more about group under investigation. They are also used for classification tasks (obviously).
Epimorphisms are properties. As long as as we have an epimorphism it means what we found a property that a group has. Iconic examples of such properties are determinants or sign of a permutation.
Monomorphism are important because they allow us to embed group under investigation into larger group, that has more structure. So we can use that structure to prove things about our current group. I think it is general principle in mathematics. E.g. proving things about real functions with complex analysis.
So my questions are:
- Does this correct/make sense?
- Can you add extra motivation/details/references (preferably iconic) about what one should "feel" about this concepts?
Your items are all interesting, and all have an element of truth to them, but I'd caution against saying that they are the only important kinds of iso/epi/mono morphisms.
Example: there is a 2-1 epimorphism of $SU(2)$ (the group of unitary 2-by-2 matrices with determinant 1) onto $SO(3)$ (the group of rotations of 3-space, aka 3-by-3 orthogonal matrices with determinant 1). This is of enormous importance in both math and physics. But if the image of $g\in SU(2)$ is $g'\in SO(3)$, I'd hesitate to say that $g'$ is a "property" of $g$ (like the determinant). Of course, "property" doesn't have a formal definition, so maybe.
Another example: if $G$ is a group, there is a canonical epimorphism to its abelianization, $G\rightarrow G^{ab}$. ($G^{ab}$ is the abelian group $G/[G,G]$ where $[G,G]$ is the subgroup of $G$ generated by all the commutators of $G$.) For any path-connected topological space $X$, the abelianization of its fundamental group is its first singular homology group. So we have an epimorphism $\pi_1(X)\rightarrow H_1(X)$. Again, I'm not sure this counts as a "property".
Isomorphism extends way beyond group theory; when two objects in a category are isomorphic they are "essentially the same thing". I guess this is what you mean by "classification".
In group representation theory, one studies homomorphisms of groups into groups of linear mappings, $G\rightarrow GL(V)$ where $V$ is a vector space and $GL(V)$ is the set of all linear 1--1 mappings of $V$ onto itself. I think this fits into your 3rd item, even though the homomorphisms need not be monomorphisms or epimorphisms. Group representation theory not only has yielded important theorems about groups, but is one of the main avenues by which group theory is applied to other fields (like physics).