Why did abstract group theory take its current form?

Question

These days, I'm interested in group theory. Why is the group axiom the way it is now? As an example, among mathematics, algebra has fundamental properties called associative property, commutative property, and distributive property. However, among these, only the associative property is included in the group axioms. And among the above properties, when the commutative property is established, it is called a special term, Abelian group. And there are certainly many good properties in the Abelian group. So I want to know why the group axiom ended up being the way it is now.

And I understand that the group theory comes from linear equations. So, in my personal opinion, I think that the generalization of possible solutions in linear equations is reflected in the group axiom, but I am asking this question because I want to know for sure. And as an additional question, what was the reason in the history of mathematics that led to thinking of groups based on linear equations?

Answer 1

The group axioms reflect the properties of bijections from a set to itself (and more generally, of automorphisms in any category), with function composition as the group operation: composition is inherently associative, and we have identity and inverse functions. In fact, every group is isomorphic to an automorphism group.

Composition isn't generally commutative. For example, rotating a shape 90 degrees around the origin doesn't commute with reflecting across the $x$-axis, so the group generated by these two operations (which is a dihedral group with 8 elements) is not abelian.

Answer 2

I can recommend reading Hans Wussing, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, see for example here.

Answer 3

I really like Socratica's Treatment of Groups for a lot of these questions. But in general, we got to group theory because it was useful to get to group theory.

You mention 3 substantially important axioms: associativity, commutativity, and distributivity. Of these, distributivity requires 2 operations, so it's really handled separately with other abstract objects. In particular, vector spaces are distributive and that gives them a lot of power!

Associativity is an interesting beast. It appears to have a very special place in the heart of mathematics. Non-associativity does exist, but its either pushed to the side or commonly we will embed the non-associative algebra in something that is associative, and then analyze that. I've tried to inquire deeper, but it does seem to be something that is informally important to mathematicians.

We do explore commutative systems. But so far we have found more application for associative systems than commutative ones. Usually commutativity is layered on later, as we see with Abelian groups.

What I think gives groups their "fundamental" feel is that, for some reason, the 3 properties of groups permit incredible power. The fact that we can fully categorize the finite simple groups is pretty marvelous. And, of course, we simply find that group patterns show up in a lot of really useful problems. We could re-invent all of the principles every time, or we can start from proving what must be true for all groups, and go from there.