How to handle group schemes by points? I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. talking about elements). Moreover, in the book Geometric Invariant Theory by Mumford, though he gave the defination of group scheme by relations of morphism, what he really used was "point"! Not the closed point, but R-point(i.e. morphism Spec(R) $\to G$, where R is a ring ).
I don't know anything about thoery on R-point, and I guess this might be the way of handling group scheme by "point". Does anyone know the corresponding theory(or reference)? And how could them hook together?
Moreover, I am also curious about how does one think of group schemes. What could go wrong if one think of group scheme by element?
This may not be a well-asked question, because of my ignorance of group schemes, and the words above is just my own feeling while reading GIT. Any comments would be great appreciated!
 A: The main point is that a group scheme does not have enough points :-)  
For example, given any number field $K$ (=finite-dimensional extension of $\mathbb Q$), Mazur-Rubin have recently proved that there exists an elliptic curve $E$ defined over  $K$ such that $E(K)$ (=points with coordinates in $K$) is the trivial group with one element: this is absurdly insufficient  for studying $E$.
More importantly the possibility to vary the field (or even scheme) and look at points of the group-scheme with coordinates in that field, i.e. the morphism point of view, is incredibly important in arithmetic geometry.  
For example, if $E$ is an elliptic curve defined over $\mathbb F_p$ it has a zeta function 
$$Z(E)= \operatorname {exp} (\sum\operatorname {card}E(\mathbb F_{p^n})\cdot\frac{T^n}{n})       
                                                            $$
which is, amazingly, a rational function in $T$ and has been the prototype of the arithmetic study of group-schemes.     
That said, over an algebraically closed field $k$ , the classical study of a group-scheme just as a group $G(k)$ is quite interesting and fruitful: it is called the geometric study of $G$, as opposed to the arithmetic study evoked above.  
Edit: points of a scheme
 Given a scheme $X$ , a point with values in a another scheme $T$ is a morphism $T\to X$. This looks rather opaque (especially if you also consider that all schemes are over another scheme $S$ !) so let me try and show that this notion is not so unreasonable as it looks.    
Consider the field (or even ring) $k$, the algebra $A=k[T_1,...,T_n]/I$ ( where $I\subset k[T_1,...,T_n]$ is an ideal) and the affine $k$-scheme  $X=Spec(A)$.
Now, what is a $T$-point of $X$ if you take for $T$ the affine scheme $T=Spec(R)$ corresponding to some $k$-algebra $R$ ?
Answer: it  just corresponds to $V(R)\subset R^n$, the set of solutions  $(r_1,...,r_n)\in R^n$ of the system of equations $$P(r_1,...,r_n)=0 \quad \text{for all} \;P(T_1,...,T_n)\in I$$  which is  the naïve interpretation of "points of $X$ with values in $R$".
And this is easy:  a $k$-morphism $Spec(R)\to X=Spec(A)$ corresponds to a morphism of $k$-algebras $\phi: A=k[T_1,...,T_n]/I\to R$ to which you just associate the naïve $n$-tuple $(r_1,...,r_n)=(\phi(\bar T_1),...,\phi(\bar T_1))\in V(R)$
