I'm in my 3rd year of a physics degree, with plans to study further in graduate school. I am currently enrolled in a group theory class, as I have heard it can be useful in many fields (particularly solid state physics, which I am interested in learning more about). However, so far all we have covered are things like the well ordering principle and some theorems about greatest common divisors. This is the course description:

Does this look like it could be useful, or would I be better off studying more applied group theory on my own?

Well, it's the basics of group theory. IMO one should know those basics for several reasons:

You won't be surprised, if groups show up, that you haven't heard of before.

You will get used to group theoretical methods which apply to your groups of interest, too.

In solid state physics especially finite groups of geometric symmetries play an important role. So the program fits.

Those guys show up nearly everywhere in physics. You can't know beforehand which you will actually need.

Actions, homomorphisms etc. apply basically to other objects as well: rings, modules, vector spaces etc. So it's good to learn them at least once in greater detail.

As a physics major, I took a course that covered the same material from the math department when I was an senior undergraduate. I never used it in graduate school, (I was a plasma physicist), but I did take a course in high-energy physics. Although group theory is used in HEP, I never found the (e.g.) theorems and proofs of sylow's theorems, characterization of small order groups or rings and (algebraic; not physical) field theory, as relevant. I also took a course in group theory applications to solid state physics between my solid state 1 and solid state 2 courses, out of Ashcroft and Mermin. Again I did not find the applications course or math algebra course to be particularly relevant to them or to each other.

In short, if you are interested in the algebraic material, you can take the course. Otherwise, pass it up. I do think an advanced linear algebra course would have been useful in upper division courses in Electrical Engineering (control theory) or advanced physics.

And you will not end up here again, and ask about the representation of / from /__ / by / in / at ##SU(2)## or ##SU(3)## in a manner that shows that you haven't the slightest idea what a representation actually is.

for that reason, i suggest looking at the algebra book by mike artin, since he illustrartes group theory with linear algebra, mentions linear groups, and introduces the basic group representation involving SU(2) and SO(3), that, at least some, physicists know and use. I.e. most introductory algebra courses in the math department omit the group representations that interest physicists and give only the abstract group theory. Your course syllabus sounds like that kind. as i recall, the book on linear algebra by evar nering included a few interesting applications at the end. of course my college course omitted those, but at least we had the book.

here is a link to the table of contents, but absurdly the listed price is almost $200, and that's for a paperback!

well at least if you learn the basic group theory you can add to it with a reading in representation theory like mike artin's treatment. i.e. just because a course does not give you everything you want, but gives you something, may not be a reason to skip it.

Very true. There is also an applied PDE course I am interested in, and it might be more useful. I think I will try and learn group theory on my own from Artin's book. Thanks for the help!

Mathematicians are interested in different questions than physicists, so unless you are interested in the questions they are (for instance, questions pertaining to number theory) you are likely wasting your time. The only thing I got from two semesters of abstract algebra was better familiarity with jargon I could have picked up with much less time and effort.

Seriously, the topics mentioned in the original post have not come up in my 25+ years of using group theoretical methods in semiconductors, the only ones that came close was the groups and group actions.