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It seems like every single textbook on group theory starts out by proving Schreier's theorem and Jordan-Holder as a consequence. Yet, I've never seen any interesting results where one would apply composition series, derived series or lower or upper central series.

My question is the following: Since Jordan-Holder, solvable and also nilpotent groups seem to be important, what do we actually gain from them? Are there any interesting proof techniques that need them? For example, a finite composition series with some properties screams some sort of an induction argument, but I've never seen a result that would use it and I can't think of anything either where I would want to use it.

For example the only thing I know of where solvability is used (hence, then name) is the standard field theory material on solvability of polynomials. Yet, I've never seen this applied anywhere else.

Can anyone enlighten me why these are presented as being so central to group theory?

eof
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  • Try to google "solvable groups", "nilpotent groups" just to have a pale idea of the ammount of books, articles and papers written on these subjects. Reading a little about Gromov's theorem on polynomial growth of finitely generated groups can be also enlightening. – DonAntonio Dec 16 '13 at 13:36
  • related questions: http://math.stackexchange.com/questions/236816/applications-of-the-jordan-holder-theorem http://math.stackexchange.com/questions/25315/how-is-a-group-made-up-of-simple-groups – spin Dec 16 '13 at 20:04

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