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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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