How to prove Renormalization Group is a Group or a Semi-Group? In physics, the Renormalization Theory is an important theory which allows to study a physical system at different scales and helps to find the fixed points of the system. 
I was trying to understand why is it called a renormalization group. While reading, I came across the argument that the scale transformation map satisfies the relation $R_{l_1l_2}[m]=R_{l_2}\circ R_{l_1}[m]$ and hence it forms a group. 
Is there any connection between the general group properties i.e. closure, associativity, identity, etc. with this above relation? I understand how to prove if a given set is a group using these properties. But I am confused how to prove that a functional map satisfies group properties.
Can someone explain how to prove it or direct me to any book or article I can refer to?
Thanks
 A: You might be overthinking it... The mathematical structure of the group is translations on a line (scales). However, in the 64 years since its introduction, the term has been broadly applied to a handful of techniques, noninvertible settings, and so you get to hear "semigroup" qualifiers tossed around. But seeking mathematical subtlety in something so primitive is almost always overkill.
The strict definition in its 1954 introduction is, 
$$
g(\mu)=G^{-1}\left(\left(\frac{\mu}{M}\right)^d G(g(M))\right)$$ 
for some invertible function G (unspecified—nowadays called Wegner's scaling function) and a constant d, in terms of the coupling g(M) at a reference scale M. 
Gell-Mann and Low realized in these results that the effective scale can be arbitrarily taken as μ, and can vary to define the theory at any other scale, by plugging in the above,
$$
g(\kappa)=G^{-1}\left(\left(\frac{\kappa}{\mu}\right)^d G(g(\mu))\right) = G^{-1}\left(\left(\frac{\kappa}{M}\right)^d G(g(M))\right).$$
Thus, group closure: as the scale μ varies, the theory presents a self-similar replica of itself, and any scale can be accessed similarly from any other scale, by group action, a formal transitive conjugacy of couplings in the solutions of Schröder's functional equation--which they apparently did not know about.
In any case, beyond group closure above, functional conjugation is associative, invertibility is (formally, not physically!) evident, and the identity is choosing μ = M .  
Lest you imagined this has something deep to do with functional conjugation, just restrict to a Wegener function chosen to be the identity, so, a line, $g(\mu)=(\mu /M)^d  g(M) $.
You might be dragged into operational tricks for defining these functions, which require enormous indirection in working out inverses, but these are mere technical issues, depending on context. If you can write such an equation or the more popular and familiar differential form thereof, you have simple line shifts. The subtleties are in the physics, not the math. 
