Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equations.
Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable[1] (note this theorem holds only in characteristic 0).
give a solvable group of Galois extensions containing the following composition factors (where
Each of the defining group actions (for example,
A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups meaning that Gj−1 is normal in Gj, such that Gj /Gj−1 is an abelian group, for j = 1, 2, ..., k. Or equivalently, if its derived series, the descending normal series where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of G. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient H/N is abelian if and only if N includes the commutator subgroup of H. The least n such that G(n) = 1 is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order.
This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order.
For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field.
The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0, Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.
More generally, all nilpotent groups are solvable.
The Feit–Thompson theorem states that every finite group of odd order is solvable.
In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.
The group S5 is not solvable — it has a composition series {E, A5, S5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to A5 and C2; and A5 is not abelian.
Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n > 4, we see that Sn is not solvable for n > 4.
This is a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals (Abel–Ruffini theorem).
This property is also used in complexity theory in the proof of Barrington's theorem.
Notice that this description gives the decomposition of
the groups of upper-triangular, or lower-triangular matrices are two of the Borel subgroups.
the Borel subgroup can be represented by matrices of the form
Any finite group whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable.
Numbers of solvable groups with order n are (start with n = 0) Orders of non-solvable groups are Solvability is closed under a number of operations.
Solvability is closed under group extension: It is also closed under wreath product: For any positive integer N, the solvable groups of derived length at most N form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (direct) products.
Burnside's theorem states that if G is a finite group of order paqb where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic.
Since a normal series has finite length by definition, uncountable groups are not supersolvable.
In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups: A group G is called virtually solvable if it has a solvable subgroup of finite index.
A solvable group is one whose derived series reaches the trivial subgroup at a finite stage.
For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes.
The first ordinal α such that G(α) = G(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group (Malcev 1949).
A finite group is p-solvable for some prime p if every factor in the composition series is a p-group or has order prime to p. A finite group is solvable iff it is p-solvable for every p. [4]