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You are not logged in. #1 20090502 21:15:58
Soluble groupsI have no doubt that Ricky and his fellow countrymen call such groups “solvable” instead. #2 20090502 22:12:17#3 20090502 22:21:00#4 20090513 20:29:52
Re: Soluble groupsSupersoluble groups are not mentioned in John F. Humphreys’s A Course in Group Theory (at least not in what I have read of the book so far) but I came across them yesterday while browsing at Foyles in Central London in a book on finite groups written by a former lecturer of mine: Prof B.A.F. Wehrfritz of Queen Mary, University of London. #5 20090513 20:57:25
Re: Soluble groupsLet’s put it another way. Hence every normal series is a subnormal series. Soluble and supersoluble groups can then be defined succinctly as follows: A group is soluble iff it has a normal series with Abelian factors. It is supersoluble iff it has a normal series with cyclic factors. NB: It can be shown that a group has a normal series with Abelian factors if and only if it has a subnormal series with Abelian factors. Hence soluble groups can be defined either way. For supersoluble groups, however, the series must be normal, not subnormal. Last edited by JaneFairfax (20090516 19:33:17) #6 20090513 22:09:59
Re: Soluble groupsWhile browsing the Web in the last few hours, I found this result: Hence a group having a (sub)normal series with nilpotent factors is also soluble. A group having a subnormal series with cyclic factors is said to be polycyclic. (This is what I learn from Wikipedia.) So every supersoluble group is polycyclic. Last edited by JaneFairfax (20090516 19:31:35) 