Prosolvable group
In mathematics, more precisely in algebra, a prosolvable 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.
