Extension problem

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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 {A_i}, where each A_i+1 is an extension of A_i 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:

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

such that there is a homomorphism 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 iff the group G is a semidirect product of K and H. Semi-direct products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from , where Aut(K) is the automorphism group of K. For a full discussion of why this is true, see semidirect product.

[edit] Classifying central extensions

A central extension is an extension

such that the image of the group K is a subgroup of the center Z(G) of G. Note that this forces K to be an abelian group.

Central extensions of H by K are classified by Ext¹(H,K), the first Ext group. For those who would prefer Ext groups of abelian groups, this is the same as Ext¹(π(H),K), where π(H) is the abelianization of H.

Proof

The first step of the proof is to assume one has such an extension, and see what can be said about it. Choose a map s from H to G that need not be a homomorphism, but that does satisfy the requirement that composing it with the quotient map q from G to H induces the identity on H. Such a map is called a fake splitting, because it would be a splitting map if it was a homomorphism.

Let F be the free group generated by the elements of H, so its elements are words of letters in H. Let R be the normal subgroup of F consisting of relations, that is, words which are trivial when considered as elements in H. These groups fit into an exact sequence

The fake splitting s produces a homomorphism from F to G by taking a word to the product of the images of the individual letters . The restriction of this map to the subgroup R will have its image entirely contained within K as a subgroup of G. This is because the image will be in the kernel of the quotient map. Therefore, a fake splitting will produce a homomorphism from R to K which will be called θ_s.

How will this homomorphism differ if a different fake splitting s' is chosen? The image of an element in H will only differ by an element of K, since the two images must be in the same coset. Therefore, one can define a homomorphism δ from F to K such that the image of any letter a is s'(a)s(a) ^{− 1}. With this established, then θ_s' = θ_s + δ, where additive notation has been used to emphasize the commutativity of homomorphisms into K.

Therefore, an extension determines a homomorphism from R to K modulo the image of homomorphisms from F to K. Furthermore, given such a homomorphism, an explicit extension can be constructed, so extensions are in one-to-one correspondence to such data. The group Hom(R,K)/Hom(F,K) is what is usually called Ext¹(H,K). To see this, apply to left-exact functor Hom(-,K) to the exact sequence of H, F and R. This gives a long exact sequence which is carried on potentially infinitely by the Ext groups, although only the first couple are relevant here.