Glossary of Group Theory - Normal Series

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 ⊲ ... ⊲ G_n = G,

i.e. any G_i is a normal subgroup of the next one G_i+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 G_i+1 / G_i, 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 g_i.

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.