Supersoluble groups are not mentioned in John F. Humphreyss 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.
Lets 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 (2009-05-15 21:33:17)
While 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 (2009-05-15 21:31:35)