[GAP Forum] Normal closure of a nilpotent subgroup

[Alan Camina]

Dera Dan

The answer is that the normal closure of a subnormal nilpotent subgroup is 
nilpotent. Essentially is is Fitting's Theorem which says that the product 
of two nilpotent normal subgroups is nilpotent. Consider the chain 
K=K_1<K_2<\cdots <K_n=G for some K nilpotent and subnormal in G, where 
each K-_{i+1} is normal  in K_{i+1}.

Note that K_2 contains the normal closure of K in K_3 but the normal 
closure of K in K_3 is the product of finite many K_3 conjugates of K all 
of which are normal in K_2. So the normal closure of K in K_3 is nilpotent 
by Fitting and note that we could replace K_2 by this normal closure and 
work our way up the chain, or use induction on the defect.

if we don't insist that the groups are finite you get locally nilpotence 
instead.

Alan

School of Mathematics
University of East Anglia
Norwich
UK NR4 7TJ

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On Thu, 26 Apr 2007, Dan Lanke wrote:

> Dear GAP Forum,
>
> I am trying to find an example of a finite group G with a subnormal nilpotent subgroup H such that the normal closure of H in G is not nilpotent. I am not
> sure if such an example exists.  I used GAP to see that no such example
> exists for |G| < 190. I run into memory problems with the computer for
> larger groups. Any suggestions please?
>
> Thanks,
> DL
>
>
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