I'm new here probably I wouldn't have a suitable way for asking suitable questions for this website really.
In group theory , I mixed between set and group in Algebra; however, I checked both of them definition.
My question here is:
Is set=group?
I think there are a large difference since we have set theory and group theory ? Can we say for example "set is finitely generated " like group ?
Best Answer
There are different ways to define what kind of object a group is, however it is never a set alone. It is usually described as a set with a binary operation (such that certain properties hold), or - to clarify the meaning of "with" - as a tuple (or pair) $(G,\circ)$ of a set and a binary operation (with properties). Often one speaks of "the group $G$" instead of "the group $(G,\circ)$", but that is an abuse of language; nevertheless it is very common if it is somehow clear what the operation has to be. When we speak of the group $\Bbb R$, we actually mean $(\Bbb R,+)$ (and not with multiplication as operation because that would not make a group)
Another way would be to say that a group is simply a model of the group axioms, which is a very different level of abstraction.
We could try to view a group not as a tuple but as a single "thing" as follows:
Definition. A group is a map $f$ with the following properties:
$\operatorname{dom}(f)=\operatorname{codom}(f)\times \operatorname{codom}(f)$
$f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\operatorname{codom}(f)$
For every $x\in \operatorname{codom}(f)$, there exists $y\in\operatorname{codom}(f)$ such that for every $z\in\operatorname{codom}(f)$, we have $f(z,f(x,y))=z$.
I have however never encountered this view and made it up on the spot for this answer. :) It would be fun to express 2. and 3. in terms of the projections of the direct product to its factors instead of with elements. :)