Editing Group extension (section)

===Extension problem===

The question of what groups <math>G</math> are extensions of <math>H</math> by <math>N</math> 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 <math>\{A_i\}</math>, where each <math>\{A_{i+1}\}</math> is an extension of <math>\{A_i\}</math> 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 would give us enough information to construct and classify all finite groups in general.