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
. Then the
Galois group , where
denotes the
algebraic closure of
, is prosolvable. This follows from the fact that, for any finite
Galois extension of
, the Galois group
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 ...
, with
cyclic of order
for some
,
cyclic of order dividing
, and
of
-power order. Therefore,
is solvable.
See also
*
Galois theory
References
{{reflist
Mathematical structures
Algebra
Number theory
Topology
Properties of groups
Topological groups