I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the axioms that motivated them defining groups.
My textbook asks you to list 'common features' of vector spaces and then later defines a set of axioms for vector spaces under addition, scalar multiplication and both, noting that the axioms under addition form an Abelian group. So are groups just a generalisation of vector spaces under any binary operation?
My main problem is that the book notes that 'the axioms may seem rather arbitrary' and links groups with vector spaces but doesn't elaborate.
When introducing groups, you are tasked to complete Cayley tables for the symmetries of an equilateral triangle and square. Then, similarly to the delivery of vector spaces, notes that the tables have common properties (Closure, Identity, inverse and association) and defines a group as a set of elements under a binary operation that has these features.
So what's so important about these 4 properties? For example, if 1 or 2 the properties were excluded form the axioms, or we added an extra few properties as axioms how would that cripple the effectiveness of groups?
Are the group axioms ever difficult to work with or do they always work, forgive the crude Littlewood analogy, like a mathematical skeleton key?
What is it about these 4 properties that make groups such a powerful tool in mathematics and physics?
My best guess is that a group is the best way to express our sense of symmetry and what is symmetric mathematically, but I would prefer some elaboration.
Answer
Yes, groups express symmetries of objects. If you want a symmetry to be a way to map an object into itself which preserves some property (say, location of vertices on a square, or distance between points in the plane,) and you want symmetries to be things that you can (a) chain together and (b) undo, then you've got a group. (As long as you also admit the trivial symmetry.) In this picture of symmetries as mappings, you don't really have to specify associativity, as it's a natural aspect of anything that looks like composition of functions (more precisely, of anything which fits into a category.)
You shouldn't think that groups are just generalizations of vector spaces. Anything that's a generalization from one case is a bad generalization! Groups are also generalizations of numbers (under addition and, in some cases, multiplication), symmetries of geometric objects (both discrete ones you mention and continuous ones,) and functions, under not only addition and multiplication but, critically, composition. An important historical example was the group of permutations of roots of a polynomial, which is used in the proof that quintic equations can't be solved in radicals and has led to huge areas of modern math.
As for loosening and strengthening some of the axioms: if you throw out inverses, you get monoids (semigroups, if you also throw out the identity.) These are extremely important objects in their own right, but they're too general to have a nice structure theory. The biggest problem is with quotients: surjective homomorphisms $G\to H$ naturally correspond to "normal subgroups" $K\subset G$, i.e. subgroups with $gKg^{-1}\subset K$ for all $g\in G$. There's no such simple way to characterize quotients $M\to N$ of monoids in terms of submonoids, because there's not necessarily a $g^{-1}$ for every $g$, so the most basic isomorphism theorems, which lay the foundation for everything you're likely to learn in elementary group theory, are no longer true. Thus we like to use groups instead of monoids when we can, and many actual things in the world come as groups, so we do so.
Strengthening the axioms does not "cripple the effectiveness" of groups so much as weakening them does. Various objects subject to axioms extending those of a group include abelian groups, vector spaces, modules, rings and algebras over rings, topological groups, Lie groups, and many more, and all of these are of great importance. But there are groups that are none of these things, (every group is in some sense topological, but we don't always want to think about the topology) so we start with just a plain group and strengthen the axioms in appropriate contexts.
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