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Extension problem

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

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 K by H, so such products as the wreath product provide further examples of extensions.

The question of what groups G are extensions of K 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 gives us a complete list of finite simple groups; so the solution to the extension problem gives 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 K by a group H if and only if there is an exact sequence:

1 → KGH → 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*.

mention Ext functor; mention that extensions are known

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