| Extension Problem |
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| group theory | |
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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 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 ''H'' by a group ''K'' if and only if there is an Exact Sequence : :
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