From: [Permission pending]
Subject: Re: Papers by Hall, Kaluznin on nilpotent groups
To: rusin@math.niu.edu (Dave Rusin)
Date: Sat, 28 Oct 1995 21:27:29 -0400 (EDT)
>
> Philip or Marshall Hall?
>
>
Philip Hall.
Here is the theorem as it was found in the book by Kargapolov
and Merzljakov:
Theorem: The stabilizer of series of subgroups of length r is a
nilpotent group of class <= (r choose 2).
The Kaluznin theorem is:
Theorem: Suppose that the series G = G0 > G1 > ... > Gr = 1
is a normal series (of length r) of the group G. Let phi
denote the stabilizer of the series, i.e. the group of all
automorphisms of G leaving the Gi invariant and acting
trivially on the factors Gi/Gi+1. Then phi is nilpotent
of cLass < r, and if we regard G, phi as subgroups of Hol G,
then [G, phi] is also nilpotent of class < r.
Again, any help would be appreciated.
Thanks,
[sig deleted -- djr]
==============================================================================
[The bulk of my response, now lost, is contained below -- djr]
==============================================================================
From: [Permission pending]
Subject: Re: Papers by Hall, Kaluznin on nilpotent groups
To: rusin@math.niu.edu (Dave Rusin)
Date: Mon, 30 Oct 1995 08:38:35 -0500 (EST)
> I'll have to look at some of my books when I get back to work on Monday,
> but two things spring to mind: (1) This looks like the kind of thing
> which would appear in Huppert's textbook "Endliche Gruppe" (now title
> E.G. I, I guess). Seems to me I read in there similar if perhaps weaker
> results -- some in Ch II (?) based only on central series results, and
> some later once low-dimensional cohomology is introduced. (2) If what you want
> is a proof rather than a bibliographic citation, we may be able to
> conjure one up. View a long series as a short exact sequence
> A -> G -> B. Your group of automorphisms preserves this presentation,
> giving a homomorphism of it to Aut(A) x Aut(B). The image is contained
> in the subgroup of chain-stabilizers. The kernel is a set of derivations
> from B to A. This is a little messy if A isn't abelian, and in
> general you have to worry about how that kernel is contained in the whole
> group of automorphisms, but something like that ought to work.
>
> dave
>
Thanks for your help Dave. I am looking for a bibliographic reference for
the two original papers. I found one of them:
On groups of automorphisms, J. reine angew. Math., Vol. 182 (1940), pp.194-204
Unfortunately, I will probably never find this paper. Our library resources
don't fully accomodate math majors. What my advisor and I are hopind to do
is use interlibrary loans to get these papers up to Sudbury Ontario or to
have someone fax them to us.
Your help is greatly appreciated, unforunately at my current level of under-
standing the above explanation still appears as somewhat of a foreign language.
(I am embarrassed to say it, but our school offers one course in group theory
and no more; fortunately I was able to find an excellect thesis advisor for
my fourth-year thesis and do some interesting reading on nilpotent groups.)
My hope is that I can find these papers on the internet in electronic form.
But after a few searches, it looks like I'll have to learn german before I
make any progress that way.
Thanks again,
[sig deleted-- djr]