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#1 2009-05-01 23:15:58

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Soluble groups



smile

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


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#2 2009-05-02 00:12:17

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Soluble groups

smile


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#3 2009-05-02 00:21:00

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Soluble groups

smile


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#4 2009-05-12 22:29:52

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Soluble groups



smile

Supersoluble 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. Clapping.gif


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#5 2009-05-12 22:57:25

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Soluble groups

Let’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. cool

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)


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#6 2009-05-13 00:09:59

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Soluble groups

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. smile

Last edited by JaneFairfax (2009-05-15 21:31:35)


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