When you are self-studying, do you just pick a topic/book which takes your fancy, or is there a sorta list you like to follow?
QFT typically works with groups rather than fields. A "field" in QFT is simply a name we (Physicists) give to something we don't know how to otherwise describe. A field appears to permeate all space at all times. Fields have energy and momentum, interact with other fields, and obey certain rigid physical laws. The structure of these laws is more or less the same, the differences being mainly the strength of the interactions and the symmetries of the group structure that apply to the field. Classical field theory started with the description of the electric field, then of the magnetic field. The concept explained how a charged particle could react to the presence of another particle without touching: action at a distance. A "quantum" field is of a similar nature.
That being said, the similarity of the terms is basically a coincidence. However I have heard of some of the more Mathematical treatments of Feynman diagrams referring to things like "vertex algebras" which my poor understanding takes to be the particular algebra that the field variables obey in specific Feynman diagrams. Does the Mathematics of the field variables create a (Mathematically defined) field? I've never heard of such, but then I can't follow the Math at that level to be able to tell you for sure.
-Dan
As it was noted before group theory is used in theoretical physics.
Lie groups and algebras are used in classical electrodynamics, quantum electrodynamics and relativity theory (Lorentz group). Group theory is also used in solid state physics for investigating symmetries of crystals.
Algebra can also have more practical application. I heard it is used in cryptography and signal processing (quaternions).
I have found also that link: Group Theory and Physics, maybe it will interest you. I don't have time for reading that because I'm at work recenty