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.

]]>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.

]]>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.

I have no doubt that Ricky and his fellow countrymen call such groups solvable instead.

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