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I need a counterexample for this fact that:

If $G$ is supersolvable, so any quotient group of it, is not neccesarily supersolvable. A infinite one is prefered.

Thanks

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    I am not really sure what your question is asking. Are you looking for a supersolvable group $G$ such that some factor group of $G$ is not supersolvable? If so, then there is no such example; it is an easy exercise that every quotient group of a supersolvable group is supersolvable.2011-06-01
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    @Amitesh: Thanks for your advice. I just read the book "infinite soluble groups by J.S.Robinson" and saw he didn't point that any factor group of supersolvable group is also supersolvable. So, I asked this simple question here.2011-06-01
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    Dear Ahilum, no problem! I suggest, if you are learning about soluble groups in general, to take a look at Martin Isaacs' "Finite Group Theory". This book is an excellent introduction to the theory of finite groups and discusses supersolvability in some of the exercises, for example. Also, a good book on the theory of *finite* solvable groups is "Finite Soluble groups" by Klaus Doerk and Trevor O. Hawkes.2011-06-01

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This is a nice page about supersolvability:

http://groupprops.subwiki.org/wiki/Supersolvable_group

As stated on that page, this property is preserved by taking quotients.