Normal Series
Most of the notions developed in group theory are designed to tackle non-abelian groups. There are several notions designed to measure how far a group is from being abelian. The commutator subgroup (or derived group) is the subgroup generated by commutators, whereas the center is the subgroup of elements that commute with every other group element.
Given a group G and a normal subgroup N of G, denoted N ⊲ G, there is an exact sequence:
- 1 → N → G → H → 1,
where 1 denotes the trivial group and H is the quotient G/N. This permits the decomposition of G into two smaller pieces. The other way round, given two groups N and H, a group G fitting into an exact sequence as above is called an extension of H by N. Given H and N there are many different group extensions G, which leads to the extension problem. There is always at least one extension, called the trivial extension, namely the direct sum 'G = N ⊕ H, but usually there are more. For example, the Klein four-group is a non-trivial extension of Z2 by Z2. This is a first glimpse of homological algebra and Ext functors.
Many properties for groups, for example being a finite group or a p-group (i.e. the order of every element is a power of p) are stable under extensions and sub- and quotient groups, i.e. if N and H have the property, then so does G and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group G does not have any (non-trivial) normal subgroups, G is called simple. The name is misleading because a simple group can in fact be very complex. An example is the monster group, whose order is about 1054. The finite simple groups are known and classified.
Repeatedly taking normal subgroups (if they exist) leads to normal series:
- 1 = G0 ⊲ G1 ⊲ ... ⊲ Gn = G,
i.e. any Gi is a normal subgroup of the next one Gi+1. A group is solvable (or soluble) if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients Gi+1 / Gi, one obtains central series which lead to nilpotent groups. They are an approximation of abelian groups in the sense that
, g3] ..., gn]=1
for all choices of group elements gi.
There may be distinct normal series for a group G. If it is impossible to refine a given series by inserting further normal subgroups, it is called composition series. By the Jordan–Hölder theorem any two composition series of a given group are equivalent.
Read more about this topic: Glossary Of Group Theory
Famous quotes containing the words normal and/or series:
“A few ideas seem to be agreed upon. Help none but those who help themselves. Educate only at schools which provide in some form for industrial education. These two points should be insisted upon. Let the normal instruction be that men must earn their own living, and that by the labor of their hands as far as may be. This is the gospel of salvation for the colored man. Let the labor not be servile, but in manly occupations like that of the carpenter, the farmer, and the blacksmith.”
—Rutherford Birchard Hayes (1822–1893)
“If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.”
—Shoshana Zuboff (b. 1951)