Prosolvable group
Introduction 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 . Then the Galois group , where  denotes the algebraic closure of , is prosolvable. This follows from the fact that, for any finite Galois extension  of , the Galois group  can be written as semidirect product , with  cyclic of order  for some ,  cyclic of order dividing , and  of -power order. Therefore,  is solvable.
See also * Galois theory
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