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solvable group

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Definition

Definition

(solvable group)
A group is solvable if it is a finite iterated group extension of an abelian group by abelian groups.

In other words, a group GG is solvable if and only if there exists a finite sequence

{1}G 1G 2G k=G, \{ 1\} \;\subset\; G_1 \;\subset\; G_2 \;\subset\; \ldots \;\subset\; G_k \;=\; G \,,

in which

  1. G j1G_{j-1} is normal in G jG_j

  2. the quotient groups G j/G j1G_j/G_{j-1} are abelian.

Solvability can equivalently be expressed in terms of breaking a group down rather than building it up.

Definition

(solvable group)
Given elements g,hg,h of a group GG, the commutator is [g,h]:=g 1h 1gh[g,h] := g^{-1} h^{-1} g h. The commutator subgroup of GG is the subgroup of GG generated by the commutators [g,h][g,h] for all g,hGg, h \in G. A group is solvable if the series of groups produced by repeatedly taking the commutator subgroup, the derived series, terminates with the trivial group after finitely many steps.

Examples

Example

(Galois groups)
The terminology “solvable groups” comes from elementary Galois theory: every polynomial equation ϕ\phi over an integral domain KK has a corresponding Galois group Gal(ϕ/K)Gal(\phi/K), and Gal(ϕ/K)Gal(\phi/K) is a solvable group if and only if ϕ\phi is a solvable equation (meaning that all its solutions in an algebraic closure of KK are expressible using the elements of KK, the field operations, and extraction of roots).

Informally, this can be thought of as each step in the derived series of the Galois group requiring an additional level of nesting of radicals. If the derived series does not terminate, then the solutions of the polynomial equation cannot be expressed by radicals, no matter how deeply nested.

Example

(nilpotent groups) A nilpotent group is a solvable group given by central group extensions.

References

Textbooks:

Articles:

  • L. Goldmakher, “Arnold’s elementary proof of the insolubility of the quintic”. pdf

See also:

Last revised on April 23, 2021 at 13:16:27. See the history of this page for a list of all contributions to it.