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

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