This may turn out to be a bit embarassing, but here it is. It is probably not true that the ascending and descending central series (\*) of a nilpotent group have the same terms. (Or at least one of MacLane-Birkhoff, Rotman and Jacobson would have mentioned it.) However, I have been unable to find an example where they are different. I thought I had a sketch of proof that they are always equal, but there is a gap, of the kind where you feel it is not patchable.


I've proved it for a few nilpotent groups (dihedral of the square, any group of order p^3, the Heisenberg groups of dimension 3 and 4 over any ring -- I think the argument extends to any dimension), and checked a few more exotic examples in the excellent [Group Properties Wiki](http://groupprops.subwiki.org/wiki/Main_Page). So,



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> What is the simplest (preferably finite) nilpotent group such that its a.c.s. and d.c.s. are different?
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and



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> Do the a.c.s. and d.c.s. coincide in some interesting general case?
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(\*) For completeness, the ascending central series of a group G is defined by Z\_0 = 1, Z\_{i+1} = the pullback of Z(G/Z\_i(G)) along the projection, and the descending central series by G\_0 = G, G\_{i+1} = [G,G\_i]. The group G is nilpotent iff ever Z\_m = G or G\_m = 1. It turns out that if such m exists it is the same for both.



Nilpotent group with ascending and descending central series different?
It's false even if $m=2$. Try the product of a group of order $2$ and a dihedral group of order $8$. The center has order $4$ and the quotient by the center is abelian. The commutator subgroup has order $2$.


