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