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