For a prime p, the p-core of a finite group is defined to be its largest normal p-subgroup.

A finite group G is said to be p-constrained for a prime p if .

This means there must be a prime p, where that divides N.

Every prime p 1 (mod 4) is the sum of two squares.

In particular, we can define rings Z for any prime p in complete analogy.

For a single prime p it may be referred to as a p-root group.

For any prime p, the extension is normal of degree p(p 1).

All groups of order p is metabelian (for prime p).

For prime p, if but , then for we have .

Again we can take a prime p from the first product A* and find out that is equal to some number in the product B*.