Books on the applications of group theory.

Question

Background:

Applications abound!

I am aware of applications of group theory in general:

• cryptography.
• physics.
• chemistry.
• virology.
• computer science.
• anywhere there's symmetry.

Outside those broad strokes, though, I have no clue. The most specific amongst those on my list is the virology one; I did a module on the applications of group theory to virology as an undergraduate.

I am aware of but have not read the following:

• "Chemical Applications of Group Theory," by Cotton.
• "Group theory and its application to physical problems," by Hamermesh.
• "Group theory in physics: An introduction," by Cornwell.
• . . .

and a Google search of "books on applications of group theory" returns a bunch more I know nothing about.

The Question:

What are some books on the applications of group theory?

Context:

I'm doing a PhD in group theory, second year, having done an MPhil in group theory. Whilst I'd like to remain in Academia, competition is so fierce and abundant, that it's wise to plan as if I won't make it there. I am, thus, having a look at what I can do with group theory, not strictly yet what jobs are available.

Extra points . . .

I would be particularly grateful for books tailored towards aspects of group theory I enjoy:

• the structure of finite groups.
• combinatorial group theory.

My favourite amongst the main branches of science is biology. I'm enthusiastic about physics. Chemistry has always been something I struggled with, though I try.

Answer 1

I recommend R.Lidl, G.Pilz, Applied Abstract Algebra

I own the original version, so I know at least what I am talking about. It is a bit wider in its range of applications since it admits more general algebraic concepts, e.g. genetic algebras, or lattice theory. However, it covers the areas you have mentioned, too. You can find specific examples, many exercises, and a lot of little theorems. It's not really on PhD level, but it has a long list of references.

Finite group theory is the basis of cryptography and error-correcting codes, which are discussed in detail. I'm afraid, combinatorics a little less. Maybe within the crystallography chapter which I haven't looked very closely to.

This was in my email today: https://terrytao.wordpress.com/2024/04/24/notes-on-the-bbt-theorem/ ... will say, Terry's blog is definitely worth having a look into.