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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
whose corresponding
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
is a solvable group.


Examples

* Let ''p'' be a prime, and denote the field of p-adic numbers, as usual, by \mathbf_p. Then the Galois group \text(\overline_p/\mathbf_p), where \overline_p denotes the algebraic closure of \mathbf_p, is prosolvable. This follows from the fact that, for any finite Galois extension L of \mathbf_p, the Galois group \text(L/\mathbf_p) can be written as
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
\text(L/\mathbf_p)=(R \rtimes Q) \rtimes P, with P cyclic of order f for some f\in\mathbf, Q cyclic of order dividing p^f-1, and R of p-power order. Therefore, \text(L/\mathbf_p) is solvable.


See also

* Galois theory


References

{{reflist Mathematical structures Algebra Number theory Topology Properties of groups Topological groups