Re: p-Lower central series, relevance?

On 23 Nov, 09:00, combar...@xxxxxxxxx wrote:
Given a group G, the p-lower central series is defined by:

P_0(G)=G;
P_{i+1}(G)=[G,P_i(G)]P_i(G).

Could anyone please tell me the relevance of this subgroups, or a good
reference. I found for example that the automorphism of a finite p-
group can be computed inductively using this groups.

Thanks.

You have a typo in your definition. It should be:

P_{i+1}(G)=[G,P_i(G)]P_i(G)^p.

So P_{i+1} is the smallest normal subgroup of G such that P_i(G)/P_{i
+1}(G) is elementary abelian and central in G/P_{i+1}.

I would guess that it arises most frequently in computational group
theory, particularly when computing with finite p-groups, where it is
often the most convenient central series for computational purposes.

There is a very efficient algorithm known as the nilpotent quotient,
or p-quotient algorithm, which takes a group G defined by a finite
presentation as input, and successively computes the terms of its
lower p-central series. This has been used, for example, to compute
the restricted Burnside groups - the most recent being R(2,7), which
has order 7^20416, and took about 1 CPU-year to construct.

As a general reference and pointer to other references, I would
suggest the book "Handbook of Computational group Theory" by D F Holt,
B Eick, and E.A. O'Brien, p.355 ff.

Derek Holt.

.

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