Prosolvable group - Alchetron, The Free Social Encyclopedia

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 usually, by Q p . Then the Galois group Gal ( Q ¯ p / Q p ) , where Q ¯ p denotes the algebraic closure of Q p , is prosolvable. This follows from the fact that, for any finite Galois extension L of Q p , the Galois group Gal ( L / Q p ) can be written as semidirect product Gal ( L / Q p ) = ( R ⋊ Q ) ⋊ P , with P cyclic of order f for some f ∈ N , Q cyclic of order dividing p f − 1 , and R of p -power order. Therefore, Gal ( L / Q p ) is solvable.

References

Prosolvable group Wikipedia

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