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...

, a pro-p group (for some prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 p) is a profinite group  such that for any 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....

  the 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...

  is a p-group

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...

. Note that, as profinite groups are compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

, the open subgroup must be of finite index

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...

, so that the discrete

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

 quotient group is finite.

Alternatively, one can define a pro-p group to be the inverse limit

Inverse limit

In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...

 of an inverse system

Inverse system

In mathematics, an inverse system in a category C is a functor from a small cofiltered category I to C. An inverse system is sometimes called a pro-object in C. The dual concept is a direct system.-The category of inverse systems:...

 of discrete finite p-groups.

The best-understood (and historically most important) class of pro-p groups is the 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 with the structure of an analytic 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....

 over such that group multiplication and inversion are both analytic functions.
The work of Lubotzky

Alexander Lubotzky

Professor Alexander Lubotzky is an Israeli academic and former politician. A former head of the Mathematics Institute at the Hebrew University of Jerusalem, he served as a member of the Knesset for The Third Way party between 1996 and 1999.-Education:...

 and Mann, combined with Lazard's solution to Hilbert's fifth problem

Hilbert's fifth problem

Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...

 over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite 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...

, i.e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements.

Examples

• The canonical example is the p-adic integers
  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...
  
  

• The group of invertible n by n matrices
  Matrix (mathematics)
  In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
  
   over has an open subgroup U consisting of all matrices congruent to the identity matrix
  Identity matrix
  In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
  
   modulo . This U is a pro-p group. In fact the p-adic analytic groups mentioned above can all be found as closed subgroups of for some integer n,
• Any finite p-group
  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...
  
  is also a pro-p-group (with respect to the constant inverse system).