Powerful p-group

# Powerful p-group

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, especially in the study of p-groups
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

and pro-p-groups, the concept of powerful p-groups plays an important role. They were introduced in , where a number of applications are given, including results on Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...

s. Powerful p-groups are used in the study of automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

s of p-groups , the solution of the restricted Burnside problem , the classification of finite p-groups via the coclass conjectures , and provided an excellent method of understanding analytic pro-p-groups .

## Formal definition

A finite p-group is called powerful if the commutator subgroup
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group....

is contained in the subgroup for odd , or if is contained in the subgroup for p=2.

## Properties of powerful p-groups

Powerful p-groups have many properties similar to abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, and thus provide a good basis for studying p-groups. Every finite p-group can be expressed as a section
Section (group theory)
In group theory a section of a group G is a group that is, or is isomorphic to, a quotient group of a subgroup of G.- Examples :Of the 26 sporadic groups, 20 are sections of the monster group, and are referred to as the "Happy Family"....

of a powerful p-group.

Powerful p-groups are also useful in the study of pro-p groups
Pro-p group
In mathematics, a pro-p group is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a p-group...

as it provides a simple means for characterising p-adic
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

analytic groups (groups that are manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s over the p-adic numbers): A finitely generated pro-p group is p-adic analytic if and only if it contains an open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

that is powerful: this is a special case of a deep result of M.Lazard (1965).

Some properties similar to abelian p-groups
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

are: if is a powerful p-group then:
• The Frattini subgroup
Frattini subgroup
In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...

of has the property
• for all That is, the group generated by th powers is precisely the set of th powers.
• If then for all
• The th entry of the lower central series of has the property for all
• Every quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

of a powerful p-group is powerful.
• The Prüfer rank
Prüfer rank
In mathematics, especially in the area of algebra known as group theory, the Prüfer rank of a pro-p group measures the size of a group in terms of the ranks of its elementary abelian sections. The rank is well behaved and helps to define analytic pro-p-groups...

of is equal to the minimal number of generators of

Some less abelian-like properties are: if is a powerful p-group then:
• is powerful.
• Subgroups of are not necessarily powerful.