TITLE: What is the correct definition of a group?
QUESTION [7 upvotes]: What is the correct definition of a group? More precisely the predicate "being a group"? According to Wikipedia 

A group is a set, G, together with an operation • (called the group law of G) that...

How should one interpret this?
$\textbf{Definition A)}\\
\quad \quad G \text{ is a set},\\
\quad \quad +:G\times G\to G \\
\langle G,+\rangle \text{ is a group} :\iff\\
\quad \quad +\text{ is asscociative},\\
\quad \quad \exists 0\in G : \forall x\in G:x+0=0+x=x \text{ and } \exists y:x+y=y+x=0 $
or
$\textbf{Definition B)}\\
\quad \quad G \text{ is a set}\\
G \text{ is a group} :\iff\\
\quad \quad \exists +:G\times G\to G:\\
\quad \quad \quad +\text{ is asscociative},\\
\quad \quad \quad \exists 0\in G : \\
\quad \quad \quad \quad\forall x\in G:x+0=0+x=x \text{ and } \exists y:x+y=y+x=0 $
And is there a separate notion of "$G$ being a group with operation $+$"?

REPLY [3 votes]: You appear to be asking whether on one hand a set is a group, if an operation with the correct properties exists, or on the other hand whether the group comprises both the set and the operation.
The correct definition is that the group is set together with the operation.