Extension problem

From Wikipedia, the free encyclopedia

In group theory, if the factor group G/K is isomorphic to H, one says that G is an extension of H by K.

To consider some examples, if G = H × K, then G is an extension of both H and K. More generally, if G is a semidirect product of K and H, then G is an extension of H by K, so such products as the wreath product provide further examples of extensions.

The question of what groups G are extensions of H is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {Ai}, where each Ai+1 is an extension of Ai by some simple group. The classification of finite simple groups would give us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general.

We can use the language of diagrams to provide a more flexible definition of extension: a group G is an extension of a group H by a group K if and only if there is an exact sequence:

1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1

where 1 denotes the trivial group with a single element. This definition is more general in that it does not require that K be a subgroup of G; instead, K is isomorphic to a normal subgroup K* of G, and H is isomorphic to G/K*.

[edit] Classifying extensions

Solving the extension problem amounts to classifying all extensions of H by K; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition.

[edit] Classifying split extensions

A split extension is an extension

1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1

such that there is a homomorphism s\colon H \rightarrow G such that going from H to G by s and then back to H by the quotient map induces the identity map on H. In this situation, it is usually said that s splits the above exact sequence.

Split extensions are very easy to classify, because the splitting lemma states that an extension is split if and only if the group G is a semidirect product of K and H. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from H\to\operatorname{Aut}(K), where Aut(K) is the automorphism group of K. For a full discussion of why this is true, see semidirect product.

[edit] See also