In Ebbinghaus' Mathematical Logic:
In this and the next section we present two simple mathematical proofs. They illustrate some of the methods of proof used by mathematicians. Guided by these examples, we raise some questions which lead us to the main topics of the book.
We begin with the proof of a theorem from group theory. We therefore require the axioms of group theory, which we now state. We use $*$ to denote the group multiplication and $e$ to denote the identity element. The axioms may then be formulated as follows:
(G1) For all $x,y,z: (x * y) * z = x * (y * z)$.
(G2) For all $x : x * e = x$.
(G3) For every $x$ there is a $y$ such that $x * y = e$.
A group is a triple $(G, *^G, e^G)$ which satisfies (Gl), (G2), and (G3). Here $G$ is a set, $e^G$ is an element of $G$, and $*^G$ is a binary function on $G$, i.e., a function defined on all ordered pairs of elements from $G$, the values of which are also elements of $G$. The variables $x, y, z$ range over elements of $G$, $*$ refers to $*^G$, and $e$ refers to $e^G$.
Is it correct that a group is a structure of a first order logic system?
In a first order logic system, the axioms are specified by its deductive system (e.g. the axioms in natural deduction).
Does "axiom" in "the axioms of group theory" mean the same as "axiom" in first order logic system? Or are they two different levels of axioms?
Are only "axioms" in first order logic system axioms?
What logical concept corresponds to "axiom" in "the axioms of group theory"?
Thanks.
On P128 of Enderton's A Mathematical Introduction to Logic
The "axioms" in group theory are a nonlogical concept, just a set of formulas from logic point of view. The axioms in a proof system for a logic system are a logical conept. So they are different concepts at two different level.