Arithmetic-to-groups principle

This survey article is about a correspondence-principle between: arithmetic and groups

The arithmetic-to-groups principle is a general idea for proceeding from constructions in number theory to constructions in group theory. Here are some simple formulations of the principle:

• Suppose we have a construct or phenomenon associated with each natural number $n$. Then, try to think of a construct associated with an arbitrary (or possibly somewhat restricted, e.g. finite, nilpotent) group such that that construct, when taken for the cyclic group of order $n$, gives the given construct that we are considering.

• Suppose we have a construct or phenomenon associated with each prime power $q=p^{r}$. Then, try to think of a construct associated with an arbitrary (or possibly somewhat restricted, e.g. finite, nilpotent) group such that that construct, when taken over the elementary Abelian group of order $q$ gives the given construct.

• Suppose we have a generic construct in number theory. Then try to think of construct associated with an arbitrary group, such that, when we take the group to be the group of integers, we get the particular number-theoretic construct.