In
mathematics, a pro-''p'' group (for some
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'') is a
profinite group such that for any
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
normal subgroup the
quotient group is a
''p''-group. Note that, as profinite groups are
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, the open subgroups are exactly the
closed subgroups of finite
index, so that the
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
quotient group is always finite.
Alternatively, one can define a pro-''p'' group to be the
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of an
inverse system
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
of discrete finite ''p''-groups.
The best-understood (and historically most important) class of pro-''p'' groups is the
''p''-adic analytic groups: groups with the structure of an analytic
manifold over
such that group multiplication and inversion are both analytic functions.
The work of
Lubotzky and Mann, combined with
Michel Lazard
Michel Paul Lazard (5 December 1924 – 15 September 1987) was a French mathematician who worked on the theory of Lie groups in the context of p-adic analysis.
Career and research
Born in Paris, Lazard studied at the University of Paris–Sorbon ...
's solution to
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 mathem ...
over the ''p''-adic numbers, shows that a pro-''p'' group is ''p''-adic analytic if and only if it has finite
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
, i.e. there exists a positive integer
such that any closed subgroup has a topological generating set with no more than
elements. More generally it was shown that a finitely generated profinite group is a compact p-adic Lie group if and only if it has an open subgroup that is a uniformly powerful pro-p-group.
The
Coclass Theorems have been proved in 1994 by A. Shalev and independently by C. R. Leedham-Green. Theorem D is one of these theorems and asserts that, for any prime number ''p'' and any positive integer ''r'', there exist only finitely many pro-''p'' groups of coclass ''r''. This finiteness result is fundamental for the classification of finite ''p''-groups by means of
directed coclass graphs.
Examples
* The canonical example is the
''p''-adic integers
::
* The group
of invertible ''n'' by ''n''
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over
has an open subgroup ''U'' consisting of all matrices congruent to the
identity matrix 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 is also a pro-''p''-group (with respect to the constant inverse system).
* Fact: A finite homomorphic image of a pro-p group is a p-group. (due to J.P. Serre)
See also
*
Residual property (mathematics) In the mathematical field of group theory, a group is residually ''X'' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''".
Formally, a group ''G'' is residually ''X'' if for every non-trivial element ' ...
*
Profinite group (See Property or Fact 5)
References
*
*
Infinite group theory
Topological groups
P-groups
Properties of groups
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