p-group

Every p-group of order at most p is regular.

Every p-group of nilpotency class strictly less than p is regular.

We are going to prove that: every finite p-group has a non-trivial center.

It is an open problem whether every non-abelian p-group G has an automorphism of order p.

Some less abelian-like properties are: if is a powerful p-group then:

Every p-group of odd order with cyclic derived subgroup is regular.

Recall that a finite group is called a p-group if its order is a power of a prime p.

Every quotient group of a powerful p-group is powerful.

Any finite p-group is also a pro-p-group (with respect to the constant inverse system).

In particular, is a p-group and is solvable.

A finite p-group G is said to be regular if any of the following equivalent , conditions are satisfied:

As a corollary, every finite p-group is nilpotent.

Every finite p-group can be expressed as a section of a powerful p-group.

Every abelian p-group is regular.

Using the class equation one can prove that the center of any non-trivial finite p-group is non-trivial.

This is the Prüfer p-group.

A p-group is regular if and only if every subgroup generated by two elements is regular.

The Prüfer p-group is divisible.

For a small non-abelian example, consider the quaternion group Q, which is a smallest non-abelian p-group.

In mathematics a p-group is called power closed if for every section of the product of powers is again a th power.

There exist easily defined groups such as the p-group which are infinite periodic groups; but the latter group cannot be finitely generated.

In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group.

In other words, the "obvious" homomorphism onto an abelian p-group is in fact the most general such homomorphism.

The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ.

This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.