In
mathematics, in the field of
group theory
In 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 ...
, especially in the study of
''p''-groups 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
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \ope ...
s. Powerful ''p''-groups are used in the study of
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 automorphis ...
s of ''p''-groups , the solution of the
restricted Burnside problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influ ...
, 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
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.
The commutator subgroup is important because it is the smallest normal ...