Question

What is the purpose of group theory, group homomorphisms, etc.?

Are they used in any practical areas, or is it mostly for pure math?

Answer 1

A lot of it might be said to be "useless" (I think someone did actually say that once).

OTOH, there is often a lot to be said for stepping back from all the detail in order to see the bare essence or characteristics of a particular set of things.

To see what I mean you could begin with rotations in 3D and look at the Group theory related to that, it might surprise you.