Talk:Nilpotent group

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if G is nilpotent of class n, then both the upper central series and lower central series repeat starting at the nth term.

Is the converse also true? AxelBoldt 18:38 Nov 7, 2002 (UTC)

Hmmm. I was trying to say that n is the same in both definitions, and so the lower series becomes E precisely when the upper series becomes G; but this is an interesting question. An easy counterexample can be constructed as follows:

Let H be nilpotent of class n, and K be a non-abelian simple group; and let G = H + K. Then [K, G] = [K, K] = K; so in the lower central series, A_n = A_n+1 = K (in fact, if B_i is the lower central series of H, then A_i = B_i + K). Similarly, let Z_i be the upper central series of G, and Y_i be the upper central series of H. Since K has trivial center, Z₁ = Y₁ = Z(H); and so Z_i = Y_i. Thus, Z_n = Z_n+1 = H; so the two series begin repeating at n, but G is not nilpotent.

If K is any perfect group, it should have trivial center; giving us a bit more general construction. Also, since the lower central series is an invariant series, if it repeats, then there must be some normal subgroup K of G which is perfect; so my guess would be that in that case, G is a semidirect product of K by H, where H is nilpotent; but I need to think about this... Chas zzz brown 09:38 Nov 8, 2002 (UTC)