Шешілмелі топтар және олардың құрамындағы абелдік факторлар

Group with subnormal series where all factors are abelian

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.

For example, the smallest Galois field extension of containing the elementgives a solvable group. It has associated field extensionsgiving a solvable group of Galois extensions containing the following composition factors:

with group action , and minimal polynomial with group action , and minimal polynomial with group action , and minimal polynomial containing the 5th roots of unity excluding with group action , and minimal polynomial
, where is the identity permutation. All of the defining group actions change a single extension while keeping all of the other extensions fixed. For example, an element of this group is the group action A general element in the group can be written as for a total of 80 elements. It is worthwhile to note that this group is not abelian itself. For example:

In fact, in this group, The solvable group is isometric to , defined using the semidirect product and direct product of the cyclic groups. In the solvable group, is not a normal subgroup.

The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series is formed by just the group itself and the trivial group. But non abelian groups may or may not be solvable.

More generally, all nilpotent groups are solvable. In particular, finite p groups are solvable, as all finite p groups are nilpotent.

In particular, the quaternion group is a solvable group given by the group extensionwhere the kernel is the subgroup generated by .

Group extensions form the prototypical examples of solvable groups. That is, if and are solvable groups, then any extensiondefines a solvable group In fact, all solvable groups can be formed from such group extensions.

A small example of a solvable, non nilpotent group is the symmetric group S3. In fact, as the smallest simple non abelian group is A5, (the alternating group of degree 5) it follows that every group with order less than 60 is 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.

Consider the subgroups of for some field Then, the group quotient can be found by taking arbitrary elements in , multiplying them together, and figuring out what structure this gives. SoNote the determinant condition on implies , hence is a subgroup (which are the matrices where ). For fixed , the linear equation implies , which is an arbitrary element in since Since we can take any matrix in and multiply it by the matrixwith , we can get a diagonal matrix in This shows the quotient group .

Notice that this description gives the decomposition of as where acts on by This implies Also, a matrix of the formcorresponds to the element in the group.

For a linear algebraic group , a Borel subgroup is defined as a subgroup which is closed, connected, and solvable in , and is a maximal possible subgroup with these properties (note the first two are topological properties). For example, in and the groups of upper triangular, or lower triangular matrices are two of the Borel subgroups. The example given above, the subgroup in , is a Borel subgroup.

In the product group the Borel subgroup can be represented by matrices of the formwhere is an upper triangular matrix and is a upper triangular matrix.

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.

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.

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.

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 .

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.