Solvable group

In group theory , a branch of mathematics , a group can be resolved if it has a subnormal series with Abelian factor groups .

To the subject

The historical origins of group theory lie, among other things, in the search for a general representation of the solutions to equations of the fifth or higher degree using iterated root expressions. An iterated root expression is understood to mean the combinations of -th roots, i.e. their sums and products as well as roots from these constructs. Such a representation is also referred to as the solution of the equation and an equation for which such a representation exists as solvable . ${\ displaystyle n}$

The systematic basis for the conditions under which such a solution is possible or not possible is developed within the framework of Galois theory . Here the solvability of an equation is traced back to a special property of the Galois group belonging to the equation . This property was therefore called the dissolvability of a group .

Definitions

The most common definition is: A group can be resolved if it has a subnormal series with Abelian factor groups. In this case the series is also called solvable. Since a factor group is then exactly abelian when the corresponding normal subgroup the commutator comprises, one can, alternatively, require that the Kommutatorreihe the group eventually leads to the one group. See also the article " Series (group theory) ".

Examples and Conclusions

In the case of finite groups, the solvability is equivalent to the existence of a subnormal series with cyclic factors of prime order. This results from the fact that, on the one hand, every subnormal series can be refined into a series with simple factors and, on the other hand, every finite, simple Abelian group has a prime number order and is therefore also cyclic. The groups of prime order form the composition factors of the finite solvable groups. As is generally the case with composition series, it is also true that the composition factors are clearly defined by the group (except for the sequence) ( Jordan-Hölder theorem ), but conversely, the isomorphism type of the group cannot generally be deduced from the composition factors. In the case of the equation solution, the cyclic groups otherwise correspond to the Galois groups of body extensions through roots of body elements.

From the definition it follows immediately that Abelian groups are resolvable. At the end of the 19th century, William Burnside was able to prove that this applies to all groups of the order ( prim), see Burnside's theorem . His conjecture that all finite groups of odd order are resolvable was proven in the 1960s by Walter Feit and John Griggs Thompson . The smallest non-resolvable group is the alternating group A ₅ with 60 elements. ${\ displaystyle p ^ {n} q ^ {m}}$ ${\ displaystyle p, q}$

The symmetric group is resolvable if and only if is. Accordingly, according to Abel-Ruffini's theorem , there are general formulas only for equations up to the fourth degree, which, apart from the basic arithmetic operations, only use root expressions. ${\ displaystyle S_ {n}}$ ${\ displaystyle n <5}$

By George Polya the word was coined: "If you can not solve a problem, then there is a simpler problem you can solve!" Was in this sense (and is) the method used to solve group-theoretic problems with great success, a claim to reduce a complex group to an assertion about the compositional factors of the group. The decisive factor here is that sufficient knowledge of the simple groups that occur can be achieved. In the case of resolvable groups, the situation is particularly favorable, since the cyclic groups with prime order can be viewed very well.

Hall's theorem

A further characterization of finite solvable groups can be obtained from Philip Hall's generalizations of the Sylow theorems . Accordingly, a finite group is solvable if and only if for every maximal divisor of the group order (i.e. every natural number that divides and is too coprime) ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle m}$ ${\ displaystyle n}$ ${\ displaystyle m}$ ${\ displaystyle n}$ ${\ displaystyle n / m}$

contains a subgroup of the order , ${\ displaystyle m}$
all subgroups of the order are conjugate to one another and ${\ displaystyle m}$
every subgroup whose order divides is contained in a subgroup of the order . ${\ displaystyle m}$ ${\ displaystyle m}$

properties

Is solvable and a subgroup of , then is also solvable.

{\ displaystyle G}

{\ displaystyle H}

{\ displaystyle G}

{\ displaystyle H}

Is solvable and a normal divisor of , then is also solvable.

{\ displaystyle G}

{\ displaystyle H}

{\ displaystyle G}

{\ displaystyle G / H}

Conversely, if a normal subgroup of and and resolved, then it is also solvable.

{\ displaystyle H}

{\ displaystyle G}

{\ displaystyle H}

{\ displaystyle G / H}

{\ displaystyle G}

Is solvable and there is a surjective homomorphism of to , then is also solvable. ${\ displaystyle G_ {1}}$ ${\ displaystyle G_ {1}}$ ${\ displaystyle G_ {2}}$ ${\ displaystyle G_ {2}}$

Are and dissolvable, so is their direct product . ${\ displaystyle G_ {1}}$ ${\ displaystyle G_ {2}}$ ${\ displaystyle G_ {1} \ times G_ {2}}$

Insoluble group

A sharper form of solvability is that of super- solvability, often also called super- solubility after the English term supersolvability . A group can be resolved if it has an invariant subnormal series whose factors are cyclic. ${\ displaystyle G}$

Metabelian group

Solvable groups that have a subnormal series of length are called Metabelian. ${\ displaystyle \ leq 2}$

literature

Thomas W. Hungerford: Algebra (= Graduate Texts in Mathematics. Vol. 73). 5th printing. Springer, New York NY a. a. 1989, ISBN 0-387-90518-9 .
Stephan Rosebrock: Illustrative group theory - a computer-oriented geometrical introduction . Springer Spectrum, Berlin 2020, ISBN 978-3-662-60786-2

Web links

Alexander von Felbert: Dissolvable groups, normal and composition series . 2007 (PDF; 431 kB).