Let K be a finite group and H be a finite simple group. (A simple group is a group with no normal subgroups other than {1} and itself, sort of like a prime number.) Then the group extension problem asks us to find all the extensions of K by H: that is, to find every finite group G such that there exists a normal subgroup K' of G isomorphic to K and the quotient group G/K' is isomorphic to H.(adsbygoogle = window.adsbygoogle || []).push({});

It's pretty easy to show that, starting with the trivial group {1}, it only takes a finite number of extensions by finite simple groups to get to literally ANY finite group. And since we recently finished classifying all the finite simple groups, if we solved the extension problem we will have classified all finite groups! Group theory would essentially be over.

So has the extension problem been completely solved yet? I'm assuming it hasn't, but you never know because it may just be that the references aren't up to date. Does anyone know what the state of research is right now?

Also, this is probably a horribly naive question, but just like the extension problem has been completely solved for the case of finite abelian groups through the use of direct products, why doesn't the use of semidirect products solve the problem for all finite groups?

Any help would be greatly appreciated.

Thank You in Advance.

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# What is the latest on the Group Extension Problem?

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