The renormalization group (RG) has little to do with ``group theory'' as it is meant mathematically. Also, there is no uniqueness to the renormalization group, so the use of ``the'' in this context is misleading. Rather, the RG is an idea that exploits the physics of systems near their critical point which leads to a procedure for finding the critical point. It also offers an explanation of universality, perhaps the closest thing there is to a proof of this concept. Finally, through the scaling hypothesis, it generates relations, called scaling relations satisfied by the critical exponents. It does not allow actual determination of specific exponents. However, given a numerical calculation or some other method of determining a small subset of exponents, the scaling relations can be used to determine the remaining exponents.
In order to see how the RG works, we will consider a specific example. Consider a square spin lattice:
which has been separated into 3 
3 blocks as shown. Consider
defining a new spin lattice from the old by application of a coarse-graining
procedure in which each 3 3 block is replaced by a single spin.
The new spin is an up spin if the majority of spins in the block is up
and down is the majority point down. The new lattice is shown below:
Such a transformation is called a block spin transformation. Near a critical point, the system will exhibit long-range ordering, hence the coarse-graining procedure should yield a new spin lattice that is statistically equivalent to the old spin lattice. If this is so, then the spin lattice is said to possess scale invariance.
What will be the Hamiltonian for the new spin lattice? To answer this, consider the partition function at h=0 for the old spin lattice using the Ising Hamiltonian as the starting point:
The block spin transformation can be expressed by defining, for each block, a transformation function:
When inserted into the expression for the partition function, T acts to
project out those configurations that are consistent with the block spin
transformation, leaving a function of only the new spin variables
, in terms of which a new partition function
can be defined. To see how this works, let the new Hamiltonian be defined
through:
That this is a consistent definition follows from the fact that
Thus, tracing both sides of the projected partition function expression
over 
yields:
which states that the partition function is preserved by the block spin transformation, hence the physical properties are also preserved.
Transformations of this type should be chosen so as to preserve the functional
form of the Hamiltonian, for if this is done, then the transformation
can be iterated in exactly the same way for each new lattice produced
by the previous iteration. The importance of being able to iterate the
procedure is that, in a truly ordered state formed at a critical point,
the iteration transformation will produce exactly the same lattice as
the previous iteration, thus signifying the existence of a critical
point. If the functional form of the Hamiltonian is preserved,
then only its parameters are affected by the transformation, so we
can think of the transformation as acting on these parameters.
If the original Hamiltonian contains parameters 
(e.g., the J coupling in Ising model), then the transformation
yields a Hamiltonian with a new set of parameters
, such that the new parameters are
functions of the old parameters
The vector function 
characterizes the transformation.
These equations are called the renormalization group equations
or renormalization group transformations. By iterating the
RG equations, it is possible to determine if a system has an
ordered phase or not and for what values of the parameters
such a phase will occur.
