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Prosolvable Group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group. Examples * Let ''p'' be a prime, and denote the field of p-adic numbers, as usual, by \mathbf_p. Then the Galois group \text(\overline_p/\mathbf_p), where \overline_p denotes the algebraic closure of \mathbf_p, is prosolvable. This follows from the fact that, for any finite Galois extension L of \mathbf_p, the Galois group \text(L/\mathbf_p) can be written as semidirect product \text(L/\mathbf_p)=(R \rtimes Q) \rtimes P, with P cyclic of order f for some f\in\mathbf, Q cyclic of order dividing p^f-1, and R of p-power order. Therefore, \text(L/\mathbf_p) is solvable. See also * Galois theory ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
