Series (group theory)

In group theory , a branch of mathematics , certain series , chains or towers of subgroups , in which each subgroup is contained in its successor (ascending series) or vice versa (descending series), of a given group G are used to investigate the structure of this group Group attributed to the study of less complex groups.

This article gives an overview of the general concept of such series. It gives the definitions of certain descending series with additional properties, the normal series , subnormal series , composition series , resolvable series and the series of the derived groups as well as the ascending central series . The connection between these series, which play an important role in investigations especially of finite groups, is explained. In addition, some classical theorems about these series, such as the Schreier theorem and the Jordan-Hölder theorem, are presented.

Each of the rows described herein is a linearly ordered part association in the association of the subgroups of G . Outside of group theory in the narrower sense, these series have application in Galois' theory of field extensions , where with a finite-dimensional Galois extension (also normal extension ) each such series in the subgroup of Galois group G corresponds to a tower of (intermediate) extension fields.

Notation and ways of speaking

The series dealt with here are mainly of interest when examining non-commutative groups , so, as is customary in this context, the connection in the group is represented as a multiplication by a point or left out (juxtaposition), the neutral element of the group as and the trivial subgroup or one group that only contains the neutral element, abbreviated as 1. ${\ displaystyle e}$

The symbols "<" and " " between subgroups denote the real subgroup or normal part relation . Is so called the number ( cardinality ) of cosets of the subgroup in . Is so called the quotient group of G by the normal subgroup N . ${\ displaystyle \ triangleleft}$ ${\ displaystyle U <G}$ ${\ displaystyle [G: U]}$ ${\ displaystyle U}$ ${\ displaystyle G}$ ${\ displaystyle N \ triangleleft G}$ ${\ displaystyle G / N}$

Definitions

A row , chain or tower of subgroups of a group G is a subset of the subgroup association linearly ordered by the subgroup relation < . This definition therefore only specializes the concept of a chain , explained in the article order relation , on the subgroup relation .

In the literature, when defining this series, a numbering of the elements is occasionally introduced, then a finite chain can be written as

{\ displaystyle 1 <\ cdots <G_ {1} <G_ {2} <\ cdots <G_ {n} <\ cdots <G}

.

With this notation, the difference must be explicitly required and descending chains ${\ displaystyle G_ {k} \ neq G_ {k + 1}}$

{\ displaystyle G> \ cdots> G_ {1}> G_ {2}> \ cdots> G_ {n}> \ cdots> 1}

require a separate definition. (In both cases, only the numbered subsets belong to the chain under consideration). Unless expressly stated otherwise, the predecessor and successor series described below always have different subgroups, even with numbering.

Descending series: resolvable, subnormal, normal and composition series

A finite (descending) chain of subgroups is called a subnormal series if every real subgroup of the chain is a normal subgroup of its predecessor, i.e. if it always holds. The factors in this series are the factor groups . If each of the subgroups is even a normal divisor of , then the chain is called normal series . Members of a subnormal series count - in generalization of the term normal divisor - to the subnormal divisors . ${\ displaystyle G = G_ {0}> G_ {1}> \ cdots> G_ {n} = \ {1 \}}$ ${\ displaystyle 1 \ leq k \ leq n}$ ${\ displaystyle G_ {k} \ triangleleft G_ {k-1}}$ ${\ displaystyle G_ {k-1} / G_ {k}}$ ${\ displaystyle G}$

In the literature, the term “normal series” is occasionally used for the chain called “subnormal series”. The terminology used here is based on Hungerford (1981).

A one-step refinement of a subnormal series is any subnormal series that is created from this chain by inserting an additional subgroup (in or at the end of the chain). A refinement is a subnormal series that results from a finite number of one-step refinements. Note that in this context refinements are always real (the chain gets longer) and the chain always remains finite. ${\ displaystyle G = G_ {0}> G_ {1}> \ cdots> G_ {n}}$

A subnormal series that descends from G to 1 is called a composition series if each of its factors is a simple group ; it is called a solvable series if each of its factors is a commutative group. ${\ displaystyle G_ {k-1} / G_ {k}}$

Two subnormal series S and T are called equivalent if there is a bijection between the factors of S and T such that the factors assigned to each other are isomorphic groups.

Series of derived groups

A special descending chain of subgroups is obtained by continuing the formation of the commutator group. The commutator group of a group is the smallest subgroup that contains all commutators from , i.e. the product ${\ displaystyle K (G)}$ ${\ displaystyle G}$ ${\ displaystyle G}$

{\ displaystyle K (G) = \ langle \ {ghg ^ {- 1} h ^ {- 1} \ mid g, h \ in G \} \ rangle}

.

The commutator group is also called the first derivative group . If one continues the formation of the commutator, one has the recursion rule . The group is then called the -th derived group of . ${\ displaystyle G ^ {(1)}}$ ${\ displaystyle G ^ {(k + 1)} = K (G ^ {(k)})}$ ${\ displaystyle G ^ {(k)}}$ ${\ displaystyle k}$ ${\ displaystyle G}$

The derived groups form a descending chain of subgroups

{\ displaystyle G = G ^ {(0)}> G ^ {(1)}> \ cdots}

,

which can become constant after a finite number of steps, with commutative groups this is already the case after one step. Since the derived groups are characteristic subgroups in , this series represents a subnormal series (even a normal series), the derived groups are even fully invariant . The factors in the series are commutative groups according to the construction of the commutator group. This normal series can therefore be resolved exactly if it descends to 1. (It is, of course, generally not a composition series, as its factors need not be simple.) ${\ displaystyle G ^ {(1)} = 1}$ ${\ displaystyle G}$

A group is called solvable if its series of derived groups descends to 1, i.e. if there is a natural number such that it holds. Detailed explanations of these groups can be found in the article " Resolvable Group ". ${\ displaystyle G}$ ${\ displaystyle n}$ ${\ displaystyle G ^ {(n)} = 1}$

Ascending central row

Be a group, then the center of the group is a normal divisor of . The archetype of the center under the canonical projection is noted as. Continuing this one comes to an ascending series of subgroups ${\ displaystyle G}$ ${\ displaystyle C_ {1} (G) = C (G)}$ ${\ displaystyle G}$ ${\ displaystyle C (G / C (G))}$ ${\ displaystyle G \ rightarrow G / C (G)}$ ${\ displaystyle C_ {2} (G)}$

{\ displaystyle 1 = C_ {0} (G) <C_ {1} (G) <C_ {2} (G) <\ dotsb}

the central ascending row of . This can be constant after a finite number of steps; for commutative groups this is the case after one step, for groups with a center 1, such as simple non-commutative groups, after step 0. A group whose central sequence rises after a finite number of steps up to the group itself, for which there is a number with which applies, is called nilpotent . These groups are described in more detail in the article " Nilpotent Group ". They can always be resolved because their central series is a resolvable normal series. ${\ displaystyle G}$ ${\ displaystyle n}$ ${\ displaystyle C_ {n} (G) = G}$

Sentences and properties for descending chains

Every finite group has a series of compositions.
Every refinement of a resolvable series is resolvable.
A subnormal series is a composition series if and only if it does not allow any (real) refinements.
A group is resolvable if and only if it has a resolvable series.
A finite group can be resolved if and only if it has a composition series whose factors are cyclic groups with prime order.
A series of compositions does not allow any (real) refinement.

Lemma von Zassenhaus (also: Butterfly Lemma or Butterfly Lemma)

Lemma von Zassenhaus (named after Hans Zassenhaus ):

Let it be subgroups of a group G and let it apply . Then: ${\ displaystyle A ^ {\ ast}, A, B ^ {\ ast}, B}$ ${\ displaystyle A ^ {\ ast} \ triangleleft A; \, B ^ {\ ast} \ triangleleft B}$

${\ displaystyle A ^ {\ ast} \ cdot (A \ cap B ^ {\ ast}) \; \ triangleleft \; A ^ {\ ast} \ cdot (A \ cap B)}$ ,
${\ displaystyle B ^ {\ ast} \ cdot (A ^ {\ ast} \ cap B) \; \ triangleleft \; B ^ {\ ast} \ cdot (A \ cap B)}$ ,
${\ displaystyle \ left (A ^ {\ ast} \ cdot (A \ cap B) \ right) / \ left (A ^ {\ ast} \ cdot (A \ cap B ^ {\ ast}) \ right) \ ; \ cong \; \ left (B ^ {\ ast} \ cdot (A \ cap B) \ right) / \ left (B ^ {\ ast} \ cdot (A ^ {\ ast} \ cap B) \ right )}$ .

This lemma can be used to refine subnormal series or normal series. It is important as a technical lemma in the proofs of the following theorems.

Theorem of Schreier

Schreier's theorem (named after Otto Schreier ): Two subnormal series (or normal series) of a group G are either equivalent or can be extended to equivalent subnormal series (or normal series) by refinement (one or both series).

comment

The sentence also means that two subnormal series or normal series of a group that cannot be refined (i.e. are maximum chains with the respective additional property) must always be equivalent.

Jordan-Hölder theorem

Theorem of Jordan-Hölder (named after Camille Jordan and Otto Hölder ): Any two composition series of a group G are equivalent. Therefore, each group that has a composition series determines a unique list of simple groups (with a unique multiplicity for each simple group).

Remarks

The proposition does not claim that there is a series of compositions for a given group.
The list of simple groups mentioned in the sentence is the list of factors in any composition series. While a subgroup can only appear once in the composition series according to the definition used here, two different factors can be isomorphic. The list determines the (always finite) multiplicity with which a certain simple group appears in the list. The order of the (isomorphism types of) factors in different composition series is neither clear nor free. There are generally series of compositions with a different order of the simple factors, but on the other hand there is not a series of compositions in which they appear in this order for every arbitrary arrangement of the (isomorphic types of) simple factors from the list.

literature

Thomas W. Hungerford: Algebra . 5. print. Springer-Verlag, 1989, ISBN 0-387-90518-9

Web links

Peer pressure IV - proof of the Butterfly Lemma (or 3rd isomorphism theorem) matheplanet.com

mathematik-netz.de: Solvable groups, normal and composition series (PDF; 431 kB) - examples and evidence.