I'm reviewing some group theory with Alperin and Bell's Groups and Representations, and they chose to title the chapter containing the Sylow theorems, $p$-groups and compositions series by Local Structure. To preface the chapter they write
"In many branches of mathematics, it is profitable to study an issue by somehow “localizing” with respect to a given prime number. In this chapter, we adapt this doctrine to group theory by studying finite groups through their subgroups of prime-power order. This notion of looking at the “local structure” of finite groups has proven to be very powerful."
In so far as I understood my first encounter with Sylow theory, we're essentially leveraging the properties of prime numbers to make claims about the properties of groups with prime power orders. Is there an intuitive way to understand why a mathematician might informally refer to this as local structure, or to localizing with respect to a prime? Or is it just because $p$-Sylow subgroups are "substructures" of a larger group $G$ and hence one can view studying the components as being a local property of $G$?