2

I have learned the concepts for solvable, supersolvable, and nilpotent groups and their associated properties. In particular, we have $$\{\mbox{nilpotent groups}\}\subset\{\mbox{supersolvable groups}\}\subset\{\mbox{solvable groups}\}$$ I know that the term "solvable" comes from Galois theory where there is a correspondence between the Galois group and the solvability by radicals. But how about "supersolvable" and "nilpotent"?

For "supersolvable", what makes it so special that we need to define such a term? Is there any special application of these groups that sets "solvable groups" apart?

For "nilpotent", it somehow sounds a strong property to me, e.g., it is a direct product of its Sylow subgroups when finite. But the name sounds a bit puzzling, this somewhat like the idea of homology?

PS: I am a researcher working in the field mainly related to finite group theory. I heard that these terms have some relation with Lie algebra, which I haven't learned. So I am hoping some gurus can lead my way:-)

Easy
  • 4,515
  • 1
    The wikipedia page on nilpotent groups has a section "explanation of term", which may be of interest to you. No such thing on the page of supersolvable groups, unfortunately. – sTertooy Jul 17 '18 at 06:19
  • @TastyRomeo, you have made the problem half solvable:-) – Easy Jul 17 '18 at 06:29
  • I think Jack Schmidt has answered my question in a very nice way: https://math.stackexchange.com/questions/618604/solvable-and-nilpotent-groups-normal-series-and-intuition?rq=1 – Easy Jul 17 '18 at 09:42

0 Answers0