Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
Motivation.
Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial formula_1 there is a tower of field extensionsformula_2such that
Example.
For example, the smallest Galois field extension of formula_10 containing the elementformula_11gives a solvable group. It has associated field extensionsformula_12giving a solvable group containing formula_13 (acting on the formula_14) and formula_15 (acting on formula_16).
Definition.
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 1 = "G"0 < "G"1 < ⋅⋅⋅ < "Gk" = "G" such that "G""j"−1 is normal in "Gj", and "Gj "/"G""j"−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. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to "n"th 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.
Examples.
Abelian groups.
The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series being given by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.
Nilpotent groups.
More generally, all nilpotent groups are solvable. In particular, finite "p"-groups are solvable, as all finite "p"-groups are nilpotent.
Quaternion groups.
In particular, the quaternion group is a solvable group given by the group extensionformula_18where formula_19 is the subgroup generated by formula_20.
Group extensions.
Group extensions form the prototypical examples of solvable groups. That is, if formula_21 and formula_22 are solvable groups, then any extensionformula_23defines a solvable group formula_24. In fact, all solvable groups can be formed from such group extensions.
Nonabelian group which is non-nilpotent.
A small example of a solvable, non-nilpotent group is the symmetric group "S"3. In fact, as the smallest simple non-abelian group is "A"5, (the alternating group of degree 5) it follows that "every" group with order less than 60 is solvable.
Finite groups of odd order.
The celebrated 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.
Non-example.
The group "S"5 is not solvable — it has a composition series {E, "A"5, "S"5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to "A"5 and "C"2; and "A"5 is not abelian. Generalizing this argument, coupled with the fact that "A""n" is a normal, maximal, non-abelian simple subgroup of "S""n" for "n" > 4, we see that "S""n" 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.
Subgroups of GL2.
Consider the subgroupsformula_25 of formula_26for some field formula_27. Then, the group quotient formula_28 can be found by taking arbitrary elements in formula_29, multiplying them together, and figuring out what structure this gives. Soformula_30Note the determinant condition on formula_31 implies formula_32, hence formula_33 is a subgroup (which are the matrices where formula_34). For fixed formula_35, the linear equation formula_36 implies formula_37, which is an arbitrary element in formula_38 since formula_39. Since we can take any matrix in formula_40 and multiply it by the matrixformula_41with formula_37, we can get a diagonal matrix in formula_40. This shows the quotient group formula_44.
Remark.
Notice that this description gives the decomposition of formula_40 as formula_46 where formula_47 acts on formula_48 by formula_49. This implies formula_50. Also, a matrix of the formformula_51corresponds to the element formula_52 in the group.
Borel subgroups.
For a linear algebraic group formula_21 its Borel subgroup is defined as a subgroup which is closed, connected, and solvable in formula_21, and it is the maximal possible subgroup with these properties (note the second two are topological properties). For example, in formula_55 and formula_56 the group of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup formula_57 in formula_58 is the Borel subgroup.
Borel subgroup in GL3.
In formula_59 there are the subgroupsformula_60Notice formula_61, hence the Borel group has the formformula_62
Borel subgroup in product of simple linear algebraic groups.
In the product group formula_63 the Borel subgroup can be represented by matrices of the formformula_64where formula_65 is an formula_66 upper triangular matrix and formula_67 is a formula_68 upper triangular matrix.
Z-groups.
Any finite group whose "p"-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.
OEIS values.
Numbers of solvable groups with order "n" are (start with "n" = 0)
Orders of non-solvable groups are
Properties.
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. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.
Burnside's theorem.
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.
Related concepts.
Supersolvable groups.
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. The alternating group "A"4 is an example of a finite solvable group that is not supersolvable.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
Virtually solvable groups.
A group "G" is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.
Hypoabelian.
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. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. 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 .