Solvable group

In group theory, a branch of mathematics, a group is solvable if it has a Subnormalreihe with abelian factor groups.

The concept

The historical origins of group theory are among other things in the search for a general representation of the solutions of equations of the fifth degree or higher by means of iterated root expressions. Under an iterated root expression refers to the combinations of n - th roots, so their sums and products, roots from these constructs. Such a representation is also referred to as the resolution of the equation and an equation for which there exists such a representation, and therefore as resolvable.

The systematic foundations for the conditions under which such a solution is possible or not possible, to be developed in the context of Galois theory. Here, the solubility of an equation is attributed to a special characteristic of the members of the Galois equation. This property is therefore referred to as the solvability of a group.

Definitions

The most common definition is: A group is solvable if it has a Subnormalreihe with abelian factor groups. In this case, also called the series itself solvable, and one can conclude that in general each Subnormalreihe the group is solvable. Since a factor group is abelian if and only if the corresponding normal subgroup includes the commutator subgroup, then you can simply demand that the continued formation of the commutator group eventually leads to one group. See also the article " series ( group theory ) ".

Examples and conclusions

For finite groups, the solvability is equivalent to the existence of a Subnormalreihe with cyclic factors of prime order. This follows from the fact that on the one hand each Subnormalreihe can be refined to a number of simple factors and on the other hand, every finite simple abelian group of prime order and hence is also cyclic. The groups of prime order thus form the composition factors of finite solvable groups. As generally in composition series also applies here is that although the composition factors are uniquely determined by the group ( except for the order ) ( set of Jordan - Hölder ), but that, conversely, can not generally be inferred from the composition factors of the isomorphism type of the group. In the case of equation resolution, the cyclic groups correspond to the rest of the Galois groups of field extensions by roots of body elements.

From the definition it follows immediately that abelian groups are solvable. End of the 19th century William Burnside was able to prove that this is true for all groups of order ( prime). His conjecture that all finite groups of odd order are solvable, it was proved in the 1960s by Walter Feit and John Griggs Thompson. The smallest resolvable group is the alternating group with 60 elements.

The symmetric group is then exactly solvable if it is. Accordingly, there are also only for equations up to the fourth degree general solution formulas that use other than the basic arithmetic operations only root phrases.

By George Polya comes the saying: " If you can not solve a problem, then there is a simpler problem that can be solved! " In this sense was ( and is ) used for solving group-theoretic problems with great success the way a claim about a complicated group to reduce to a claim about the composition factors of the group. Here it is crucial that a sufficient knowledge of the occurring simple groups can be achieved. In the case of solvable groups, the situation is particularly favorable since the cyclic groups of prime order can be remarkably well overlooked.

Hall's theorem

Further characterization of finite solvable groups are obtained from the coming of Philip Hall generalizations of Sylow phrases. Thus a group G is solvable if for each maximal divisor of the group order (ie any natural number that divides and relatively prime )

• Contains a subgroup of order,
• All subgroups of order are conjugate to each other and
• Each subgroup divides the order of which is contained in a subgroup of order.

Properties

• If G is solvable and H is a subgroup of G, then H is solvable.
• If G is solvable and H is a normal subgroup of G, then it is also solvable.
• If G is solvable and there is a homomorphism from G to H, then H is solvable.
• Are H and resolved, including G
• If G and H be resolved, as well as their direct product.

Superauflösbare group

A sharper form of solvability is that of Superauflösbarkeit, often called Überauflösbarkeit. A group G is superauflösbar if it has an invariant Subnormalreihe whose factors are cyclic.