Talk:Nilpotent group

Point of stabilization of central series

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)

p-groups have non-trivial center??

Also when infinite? —Preceding unsigned comment added by 81.210.238.84 (talk) 07:34, 15 April 2008 (UTC)

No, see Tarski monster group. --Zundark (talk) 09:33, 15 April 2008 (UTC)

Adjoint action

The definition of adjoint action in the article is logically self-contained but links to a different (if related) concept in Lie algebras. This seems unhelpful. Quotient group (talk) 19:33, 9 October 2009 (UTC)

Rating

I have inserted a math-rating above, because I think that this important topic deserves further expansion. Many articles on special classes of groups are better developed, even if their subject-matter is (arguably) mathematically less vital and less widely applicable. Other views?

Alternatively, ought this page to be merged with the nice page on Central Series? I rather think so.

Ambrose H. Field (talk) 19:27, 25 February 2011 (UTC)

explanation of term

Robert B. Warfield, Jr. says in his 'Nilpotent Groups', (Springer-Verlag Lecture Notes in Mathematics 513), page v: "I do not know when it was noticed that these groups are related to those connected linear Lie groups whose Lie algebras consist of nilpotent matrices, but this was certainly understood in the 1930's."

I think this is relevant to the explantion of the term 'nilpotent group.'198.189.194.129 (talk) 21:31, 4 September 2012 (UTC)

Proof of properties for finite groups

The proof for (e)→(a) is a bit strange: it seems to prove (c), and then refers to (c)→(e). But how does that help us to get to (a)? — Preceding unsigned comment added by Nomeata (talk • contribs) 12:07, 19 January 2022 (UTC)