As a ugrad math major, I had some group theory. At the end, I wrote my honors paper on group representations in part to respond to a query from the chemistry department.

I got the first dose of group theory in a standard course in abstract algebra that also covered Galois theory, rings, integral domains, fields, vector spaces, even a little on quaternions.

In grad school I was forced into another course in abstract algebra, this time from the book I. Herstein, Topics in Algebra. Otherwise the prof was trying to understand algebraic geometry, as in Grothendieck. After his first few lectures on group theory, after class I asked if he was going to cover group representations, and he was intimidated and responded "That's deep". I explained that I'd just written my ugrad honors paper on that subject. So, I didn't bother to show up for his class, and at the end he gave me a little oral exam in Galois theory, which I had to review at the time.

One good thing from this prof: He was Italian; the university was in a small town awash in dimly lit, romantic pizza shops; and some of us students asked him which pizza was better, Italian or American. He hesitated and confessed "American". So, he was both a mathematician and a diplomat!

Apparently some people think
that Herstein's book is still super good stuff: At

https://www.alibris.com

they want $36 for a used one and $200+ in some cases.

I suspect that

S. Lang, Algebra is excellent although maybe not as a first or only source. I found a PDF at
https://math24.files.wordpress.com/2013/02/algebra-serge-lan...

Just the basics of group theory are so well known, written about so often that finding good sources should be easy. Say, just go to the section of any research library, see where Herstein's or Lang's books are, and see what else is there.

Of course, here with Herstein and Lang I'm talking math, legitimately pure math, and not computing, computer science, or much if anything in recent applications.

The definition of a group and the early results are not just simple -- they are childishly, dirt simple, so simple that maybe an adult should be embarrassed to study them.

After some decades, here from just memory I will guess at the definition: A group is a non-empty set G together with an operation (here I omit a picky definition of an operation), say, + such that

(i) For any elements a, b in G, a + b is in G (likely redundant given a careful definition of an operation on G).

(ii) There is an element 0, the identity element, in G such that for any a in G we have

a + 0 = 0 + a = a

(iii) For any element a in G there exists an inverse of a, -a, such that

a + -a = -a + a = 0

(iv) The operation + is associative, that is, for any elements a, b, c in G, we have

(a + b) + c = a + (b + c)

So, let's check: Suppose a, b are in G and we have

a + b = 0

Then

-a + (a + b) = -a

Using associativity we have

(-a + a) + b = -a

Using the identity 0 we have

0 + b = -a

and finally

b = -a

and similarly for

b + a = 0

So, the inverse of a is unique.

How 'bout that! We have proved our first theorem in group theory!

For a little lesson, note the extreme care here where we use associativity and the identity. In a lot of math, we get just to slop through such details, but in group theory we have to be picky in the extreme.

To continue with group theory, we make another definition: Given group (G,+) (there is a style of math that likes such sparse, precise notation), if for any elements a, b in G, we have

a + b = b + a

then the group is commutative and Abelian (after the mathematician Abel).

Not all groups are Abelian, that is, they are not all commutative. For a simple example, given quickly (I'm omitting some picky details) in matrix theory in linear algebra matrix addition is commutative but matrix multiplication is not.

The topic group representations replaces each element of group G with (usually, maybe always) a matrix and the operation with matrix multiplication. An advantage is that we get to write down the matrices and multiply them; otherwise we are likely stuck with a big group multiplication (or in the case of calling the operation +, addition) table.

From all I can tell, the applications of group theory in quantum mechanics, particle physics, and chemistry make use of only the simplest parts of group theory.

I have a bias: In math, first I want to see an application. That is, a recipe for rabbit stew starts out "First catch a rabbit." Then maybe I can get interested in learning some math or doing some research in math that I can use for the application. I confess: This is a radical approach to math. This approach was easy enough to anticipate for someone early in their career long flooded with math topics and trying to make money to support a wife, needing to be selective, etc.

With this approach, I asked what were the applications of abstract algebra -- groups, rings, fields, etc. Not getting a good answer, I concluded that abstract algebra was abstract nonsense, tricky, picky stuff created for no good reason and returned to math analysis and its applications to physics, engineering, etc.

Well, surprising or not, that conclusion is wrong: Whatever Galois, Abel, etc. did/did not know about applications, abstract algebra got applied, has some applications. And abstract algebra is soon not childishly simple and, instead, has some deep/difficult questions and by now some deep/difficult answers.

To continue on a little, a group G might have only finitely many elements or infinitely many elements. For a group with infinitely many elements -- super easy, with R the set of real numbers, (R,+) is an example.

Given a group (G,+), maybe there is a proper subset H of G so that (H,+) (to be really picky, there is a slight abuse of notation here since an operation + cannot be the same on both G and H -- group theory is a picky subject) is also a group. Then (H,+) is a sub-group of (G,+).

Exercise: Fine a proper subgroup of (R,+) (uh, to be picky, proper here means not all of R).

So, if we let |G| denote the number of elements in a set, we have a theorem: In case G is finite, |H| factors |G| where factors means in the third grade sense divides with 0 remainder. Exercise: Prove it!

Some of the advanced results in group theory (Sylow's theorems, Jordan-Holder theory, and much more) are astounding -- tough to believe that they could be true, but they are.