Meaning of the word "axiom"
One usually describes an axiom to be a proposition regarded as self-evidently true without proof.
Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises to infer conclusions, which are called "theorems" of this theory.
For example, we can use the Peano axioms to prove theorems of arithmetic.
This is one meaning of the word "axiom". But I recognized that the word "axiom" is also used in quite different contexts.
For example, a group is defined to be an algebraic structure consisting of a set G, an operation , an element and a mapping such that the following conditions, the so-called group axioms, are satisfied:
Why are these conditions (that an algebraic structure has to satisfy to be called a group) called axioms? What have these conditions to do with the word "axiom" in the sense specified above? I am really asking about this modern use of the word "axiom" in mathematical jargon. It would be very interesting to see how the modern use of the word "axiom" historically developed from the original meaning.
Now, let me give more details why it appears to me that the word is being used in two different meanings:
As peter.petrov did, one can argue that group theory is about the conclusions one can draw from the group axioms just as arithmetic is about the conclusions one can draw from the Peano axioms. But in my opinion there is a big difference: while arithmetic is really about natural numbers, the successor operation, addition, multiplication and the "less than" relation, group theory is not just about group elements, the group operation, the identity element and the inverse function. Group theory is rather about models of the group axioms. Thus: The axioms of group theory are not the group axioms, the axioms of group theory are the axioms of set theory.
Theorems of arithmetic can be formalized as sentences over the signature (a. k. a. language) , while theorems of group theory cannot always be formalized as sentences over the signature . Let me give an example: A typical theorem of arithmetic is the case n=4 of Fermat's last theorem. It can be formalized as follows over the signature :
A typical theorem of group theory is Lagrange's theorem which states that for any finite group G, the order of every subgroup H of G divides the order of G. I think that one cannot formalize this theorem as a sentence over the "group theoretic" signature ; or can one?