Subgroup series

In group theory, a branch of mathematics, certain rows, chains or even towers of subgroups, where each subgroup is contained in its successor (ascending series) or vice versa ( descending series), a given group G are used to the structure study of these group attributable to the study of less complex groups.

This article gives an overview of the general concept of such series. He gives the definitions of certain descending order with additional properties of the normal range, Subnormalreihe, composition series resolvable row and the number of the derived groups and the ascending central series. The relationship between these rows, mainly of finite groups play a major role in investigations is explained. In addition, some classical theorems on these series as the set of Schreier and the set of Jordan - Hölder.

Each of the series described here is a linearly ordered sublattice of the Association of subgroups of G. Outside of group theory in the strict sense have these series application in the Galois theory of field extensions, where at a finite Galois extension ( also normal extension ) each such series in subgroup Association of Galois G corresponds to a tower of ( intermediate) extension fields.

• Lemma 3.1 of Zassenhaus (also: Butterfly Lemma or Schmetterlingslemma )
• 3.2 set of Schreier
• 3.3 set of Jordan - Hölder

Notation and ways of speaking

Discussed in this series are mainly in the study of non-commutative groups of interest, therefore the link in the group, as in this context usually represented as multiplication by a point or omitted ( juxtaposition ), the neutral element of the group as and trivial subgroup or one group that contains only the identity element, in short as 1

The symbols " <" and " " between sub-groups refer to the sub-group or the normal divider ratio. Is so called the number ( cardinality ) of the cosets of the subgroup in. Is so called the factor group of G by the normal subgroup N.

Definitions

A number, string, or a tower of subgroups of a group G is a linearly ordered by the subset relation < subset of the sub-group association. Thus, this definition specializes only the stated in the article order relation notion of a chain on the subset relation.

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

In this notation, the difference must be specifically requested and descending chains

Require a separate definition. ( In both cases, include only the numbered subsets of the considered chain). Unless specifically otherwise stated, with the series described in the following predecessor and successor are also available with numbering always different subgroups.

Descending series: dissolvable, Subnormal, normal and composition series

A finite (descending ) chain of subgroups is called Subnormalreihe if every proper subgroup of the chain is a normal subgroup of its predecessor, if so applies always. The factors of this series are the factor groups. If any of the subgroups even a normal subgroup of G, then that means the chain normal range. Members of a Subnormalreihe count - a generalization of the concept of normal subgroups - the Subnormalteilern.

In the literature, the term " normal range " also occasionally for here " Subnormalreihe " said chain used. The language regime as used herein is governed by Hungerford (1981).

A one-step refinement of a Subnormalreihe is any Subnormalreihe, by inserting an additional subgroup results from this chain ( in or at the end of the chain ). A refinement is a Subnormalreihe that arises from a finite number of one-step refinements. Note that in this context refinements are always real ( the chain is longer) and the chain always remains finite.

A Subnormalreihe, which descends from G to 1, ie composition series if each of its factors is a simple group, it is called resolvable row, if each of its factors is a commutative group.

Two Subnormalreihen S and T are called equivalent if there is a bijection between the factors of S and T, so that the mutually associated factors are isomorphic groups.

Number of groups derived

A special descending chain of subgroups is obtained through continuing education of the commutator subgroup. The commutator of a Group G is the smallest sub-group that contains all the commutators of G, so the product

The commutator is also referred to as the first group derived. Substituting the Kommutatorbildung continues, then one has the recursion. The group then called the k derived group of G.

The derived groups form a strictly descending chain of subgroups

After a finite number of steps can be constant, wherein the commutative groups this is after a step of the case. Since the derived groups are even characteristic subgroups in G, this series provides a Subnormalreihe ( even a normal row) represents the factors of the number are on the construction of the commutator subgroup commutative groups. This normal range is then exactly solvable if it descends to 1. ( It is of course generally no composition series, as their factors need not be simple. )

A group G is called solvable if its derived series of groups to one descends, so if a natural number n such that the following holds. Detailed explanations of these groups can be found in the article " Resolvable group ".

Ascending central series

Let G be a group, then the center of the group is a normal subgroup of G. The inverse image of the center under the canonical projection is as quoted. Substituting this further on, we come to an ascending series of subgroups

The ascending central series of G. This can be constant after finitely many steps, for commutative groups is after one step, for groups with center 1, such as simple non - commutative groups after step 0 the case. A group whose central row to the group rises even after finitely many steps, ie for which a number n such that goes with the words, nilpotent. These groups are described in more detail in the article " nilpotent group." You are always solvable, since their lower central series is a solvable normal range.

Sets and properties for descending chains

Zassenhaus lemma (also: Butterfly Lemma or Schmetterlingslemma )

This lemma can be used to refine Subnormalreihen or normal rows. It is a technical lemma in the proofs of the following sets of meanings:

Be subgroups of a group G and suppose. Then:

Set of Schreier

Two Subnormalreihen ( or normal rows ) of a group G are either equivalent or can be extended by refinement ( one or two rows ) to equivalent Subnormalreihen ( or normal rows ).

The theorem states also that two Subnormalreihen or normal series of a group, which can not be refine (ie maximal chains with the respective additional property are ), must always be equivalent.

Set of Jordan - Hölder

Any two composition series of a group G are equivalent. Therefore, each group has a composition series determines a unique list of simple groups ( with a unique multiplicity for each simple group ).