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## #1 2009-05-02 21:15:58

JaneFairfax
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### Soluble groups

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

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

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## #2 2009-05-02 22:12:17

JaneFairfax
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### Re: Soluble groups

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

## #3 2009-05-02 22:21:00

JaneFairfax
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### Re: Soluble groups

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

## #4 2009-05-13 20:29:52

JaneFairfax
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### Re: Soluble groups

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.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

## #5 2009-05-13 20:57:25

JaneFairfax
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### 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.

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 (2009-05-16 19:33:17)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

## #6 2009-05-13 22:09:59

JaneFairfax
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### 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.

Last edited by JaneFairfax (2009-05-16 19:31:35)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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