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Dear Vahid, and GAP Forum

The answer to your question is contained in the papers "Stably

soluble and stably nilpotent groups" by D. I. Zaicev (Dokl Akad

Nauk SSSR 176(1967) and Stably nilpotent groups (same author,

Mat. Zametki 2(1967) 337-346): Any infinite non-Chernikov group of

class exactly c contains a proper infinite subgroup of class c. (Of

course this is much more general than you want).

Martyn Dixon

Send reply to: GAP Forum <GAP-Forum-Reply@dcs.st-and.ac.uk>

Date sent: Thu, 06 Dec 2001 17:37:51 +0100

From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de>

To: Multiple recipients of list <GAP-Forum@dcs.st-and.ac.uk>

Subject: Re: Nilpotency class

Dear Vahid Dabbaghian, dear GAP Forum,

Last Wednesday you asked in the Forum:

If G is an infinite ( finitely generated ) nilpotent group of class n, what

information does exist about the nilpotency class of its maximal subgroups?

I do appreciate if inform me any article or paper about it.The answer is very simple: A maximal subgroup M of a nilpotent group G

of class n is of class less or equal to n. This is trivially seen

since the intersections of the groups of a central series of G with M

yield a central series of M (in which certain of the factors can

become trivial).The example of the direct product of a dihedral group of Order 2^(n+1)

with an infinite cyclic group (this direct product is of class n)

shows that there are are maximal subgroups of class n (take the direct

product of the dihedral group with the subgroup of index 2 in the

infinite cycle), as well as of class n-1 (take the direct product of

one of the two dihedral subgroups of order 2^n in the dihedraol factor

with the infinite cycle), but even of class 1 (take the direct product

of the cyclic subgroup of order 2^n in the dihedral factor with the

infinite cycle).Since this is so easy, I have never seen it stated as a theorem in

writing.Hope this answers your question.

Joachim Neubueser

************************************ Martyn Dixon Department of Mathematics University of Alabama Tuscaloosa AL 35487-0350 U.S.A e-mail: mdixon@gp.as.ua.edu phone: 205-348-5154 Fax: 205-348-7067 http://www.math.ua.edu/ http://www.bama.ua.edu/~mdixon

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