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 [math]\displaystyle{ \mathbf{Q}_p }[/math]. Then the Galois group [math]\displaystyle{ \text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p) }[/math], where [math]\displaystyle{ \overline{\mathbf{Q}}_p }[/math] denotes the algebraic closure of [math]\displaystyle{ \mathbf{Q}_p }[/math], is prosolvable. This follows from the fact that, for any finite Galois extension [math]\displaystyle{ L }[/math] of [math]\displaystyle{ \mathbf{Q}_p }[/math], the Galois group [math]\displaystyle{ \text{Gal}(L/\mathbf{Q}_p) }[/math] can be written as semidirect product [math]\displaystyle{ \text{Gal}(L/\mathbf{Q}_p)=(R \rtimes Q) \rtimes P }[/math], with [math]\displaystyle{ P }[/math] cyclic of order [math]\displaystyle{ f }[/math] for some [math]\displaystyle{ f\in\mathbf{N} }[/math], [math]\displaystyle{ Q }[/math] cyclic of order dividing [math]\displaystyle{ p^f-1 }[/math], and [math]\displaystyle{ R }[/math] of [math]\displaystyle{ p }[/math]-power order. Therefore, [math]\displaystyle{ \text{Gal}(L/\mathbf{Q}_p) }[/math] is solvable.

See also

• Galois theory