Almost. The name "group" in renormalization group was inspired by the groups in group theory, but in reality the renormalization "group" isn't a group but a sem

Almost. The name "group" in renormalization group was inspired by the groups in group theory, but in reality the renormalization "group" isn't a group but a semi-group. A semi-group satisfies the same axioms as a group except for the existence of inverse. And that is the mathematical reason why you can describe big things in terms of smaller things, but you can't describe small things in terms of bigger things: the renormalization (semi-) group flows from the ultraviolet to the infrared, but not the other way around :-)