Prosolvable group explained

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} . Then the Galois group Gal (\overline{ Q }_p/\mathbf_p), where \overline{ 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} \rtimes Q) \rtimes P , with P cyclic of order f for some f\in N , Q cyclic of order dividing p ^{f-1} , and R of p -power order. Therefore, Gal (L/ Q _{p)} is solvable.

See also

• Galois theory