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P-adic Analytic Group
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open set, open normal subgroup N\triangleleft G the quotient group G/N is a p-group, ''p''-group. Note that, as profinite groups are compact space, compact, the open subgroups are exactly the closed set, closed subgroups of finite index of a subgroup, index, so that the discrete space, discrete quotient group is always finite. Alternatively, one can define a pro-''p'' group to be the inverse limit of an inverse system of discrete finite ''p''-groups. The best-understood (and historically most important) class of pro-''p'' groups is the p-adic number, ''p''-adic analytic groups: groups with the structure of an analytic manifold over \mathbb_p such that group multiplication and inversion are both analytic functions. The work of Alexander Lubotzky, Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the ''p''-adic numbers, shows that a pro-''p' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
