Why are the group axioms called "axioms"?

Question

My book says:

A group is an ordered pair $(G, \cdot)$ where $G$ is a set and $\cdot$ is a binary operation satisfying the following axioms [emphasis mine]:

$\vdots$

Why doesn't it say

A group is an ordered pair $(G, \cdot)$ where $G$ is a set and $\cdot$ is a binary operation satisfying the following propreties:

$\vdots$

As far as I know, axioms are statements which we take for granted; I don't understand why the group axioms are something we have to take for granted; we already know that there exist objects (such as the set of permutations of $3$ objects) which satisfy those propreties.

Even if we knew of no objects which satisfy those propreties, then perhaps the axiom should be "There exists an object which satisfies those propreties" rather the axioms being the propreties themselves.

Answer 1

These axioms are not the axioms for the subject of group theory — they are axioms for the theory of a group.

Anything (in which we have interpreted the language of a group) that is a group will satisfies all of the theorems of the theory of a group. Conversely, anything satisfying all of the theorems of the theory of a group we shall call a group.

Axioms are not nearly so deep as people think they are; they are simply a way to describe and work with theories. For example, anything satisfying this list of axioms for the theory of a group will satisfy all of the theorems of the theory of a group.

The way groups are formulated is an application of model theory — formally we specify a language including a binary operation, and then we define the theory of a group by listing axioms that the binary operation must satisfy.

Then, a group is a model of this theory — we provide a set whose elements serve as the objects and a function that serves as the binary operation such that the theorems of the theory of a group are true for this set and function. (it's enough to just check the axioms)

There is actually a lot of value of taking this model theoretic perspective in abstract algebra. If we fix the traditional definition of a group by adding a unary operation (inversion) and a constant (the unit), then the theory of a group can be seen to be an algebraic theory: it can be axiomatized simply by writing a list of identities.

Algebraic theories are the topic of universal algebra which proves universal theorems that apply to a wide swath of different kinds of algebraic theories; for example, the isomorphism theorems for groups actually apply to any variety of universal algebra.

Answer 2

Axioms are properties too.

Progress on some of the ancient conjectures of mathematics became clarified by this realization.

For example, the question of whether the Parallel Postulate followed from the rest of Euclid's Axioms was settled in the negative by construction of an alternate geometry, the Hyperbolic Plane, and by then proving that the parallel postulate is a false property of the hyperbolic plane, whereas each of the remaining Euclid axioms is a true property of the hyperbolic plane.

For a simpler example, think of Peano's Axioms:

1. Each $n$ has a unique successor $s(n)$.
2. If $m \ne n$ then $s(m) \ne s(n)$.
3. There exists an element $1$ such that $n$ is a successor of some other element if and only if $n \ne 1$.
4. For each statement $P(n)$, if $P(1)$ is true and if $P(n) \implies P(n+1)$ then $P(n)$ is true for all $n$.

We can think of these as properties, and we can examine mathematical structures which satisfy these properties, or some of these properties.

For example, the integers $\mathbb{Z}$ with successor function $s(n)=n+1$ satisfies 1 and 2 but not 3 and 4.

Also, the union of the natural numbers $\mathbb{N}=\{1,2,3,\ldots\}$ with the set of all half-integers $\{...,-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2},...\}$, and with successor function $s(n)=n+1$, satisfies 1, 2, and 3 but not 4.

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Perhaps one could reconfigure your question to say something like this: When would we call a "property" an "axiom", and when would we call an "axiom" a "property"?

Roughly speaking, when we want to simply accept the existence of mathematical structures satisfying given properties, and when we then want to derive further theorems about such structures using only logical arguments starting from those given properties, then it is fair to call them "axioms". For instance, the beginnings of most group theory texts have a lot of this.

On the other hand, when we have a given set of axioms, and we want to study particular examples of mathematical structures to verify whether they do or do not satisfy those axioms, then it is fair to call them "properties". Every good group theory textbook that is worth its salt has lots and lots of actual examples of particular groups, as well as counterexamples where some of the group axioms fail, together with proofs that the axioms are or are not satisfied for those various examples and counterexamples. In this case we would say that the axioms are "properties" of those things that are constructed, properties which may or may not be true.

Answer 3

Quoting from Wikipedia: “An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments”. Don't you agree that group axioms are starting points?

Answer 4

"Axioms" in geometry were thought to be self-evident propositions expressing properties of physical space, and were therefore called "axioms".

But then it was discovered that there are spaces satisfying those axioms and other spaces not satisfying them. And so it is with groups. "Axiom", as used in mathematics, ceased to mean a self-evident proposition and took on a different meaning.

Answer 5

In my opinion, these are definitions, not axioms. Hence properties is a better word. As I see it:

Definition = gives a concise name to a complex object. It normally follows the pattern: "genus proximum et differentia specifica", as in you example of the definition of a group.

Axiom = gives a fact about a defined world, which fact cannot be proven directly from the given definitions or from the other axioms.