I am looking for good, detailed references for "mod $p$ lower central series".

So far I only find papers such as (https://core.ac.uk/download/pdf/81193793.pdf, https://www.sciencedirect.com/science/article/pii/0040938366900243), which briefly mention it in the context of topology.

Are there any good books that discuss this in detail (not necessarily related to topology)?

Also, just to confirm, are these terminologies the same thing:

  1. mod $p$ lower central series
  2. lower $p$-central series
  3. lower exponent-p central series

I am confused by the different terminologies.

Thanks a lot.


This is a fundamental in the theory of pro-$p$ groups. A good reference is

J. D. Dixon, M. Du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups, second edition, Cambridge Studies in Advanced Mathematics, 61, Cambridge University Press, Cambridge, 1999.

  • $\begingroup$ And is it the same as the lower $p$-central series? I had never heard of the mod $p$ lower central series. $\endgroup$ – Derek Holt Aug 27 '18 at 18:03
  • $\begingroup$ @DerekHolt: I would assume so. There are two possibilities, depending on whether you want the associated graded to be a restricted Lie algebra over $\mathbb{F}_p$ or only an ordinary Lie algebra over $\mathbb{F}_p$. In most situations, you want it to be a restricted one (and that is what is covered in the reference I gave). $\endgroup$ – Andy Putman Aug 27 '18 at 19:20

For the free group, it is called the Zassenahus filtration. Golod-Shafarevich groups are defined in terms of it. Ershov's survey on Golod-Shafarevich groups is excellent. Highly recommended. It is published as Ershov, Mikhail Golod-Shafarevich groups: a survey. Internat. J. Algebra Comput. 22 (2012), no. 5, 1230001, 68 pp


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