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Editorial Reviews. Book Description. Three chapters introduce readers to strong approximation Lectures on Profinite Topics in Group Theory (London Mathematical Society Each is accessible to beginning graduate students in group theory and will Language: English; ASIN: BU10; Text-to-Speech: Enabled.
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Section 10 provides a taste of current research on complex irreducible representations of compact p-adic Lie groups. It introduces the Kirillov orbit method and illustrates its use in the study of representation zeta functions. References The following books, which can be regarded as our main references, cover some of the selected material in greater detail.
They also address many related and more advanced topics. Segal, Analytic pro-p groups, Cambridge University Press, Compact p-adic Lie groups G. Klaas, C. Leedham-Green and W.
Leedham-Green and S. Throughout the text I have aimed to give reasonably complete, but not exhaustive references to the literature. A guiding principal for my choices has been to select economically a mixture of classical and modern references which are suitable for a newcomer to the subject. More complete references can be found in the books listed above. Each section of the present notes, except for the short Section 3, ends with a few selected suggestions for further reading.
I also included key results from selected research articles and preprints. Originality I can claim, in a limited sense, with regard to the overall exposition. I am grateful to Dan Segal, Christopher Voll and the anonymous referees for their comments on earlier versions of this text. Conjugation provides a natural action of G on itself; indeed, it induces a homomorphism from G into its automorphism group Aut G.
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The kernel of this homomorphism, which constitutes a normal subgroup of G, is called the centre 2. The subgroup generated by all commutators is called the commutator subgroup of G and denoted by [G, G]. The group [G, G] can be characterised as the smallest normal subgroup of G such that the corresponding quotient is abelian.
Nilpotent groups can be thought of as close relatives of abelian groups. This fact can easily be proved 12 Chapter I. Compact p-adic Lie groups inductively from the following fundamental observation.
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This observation can be proved by analysing the possible orbit sizes in the action of G on N by conjugation; see Exercise 4. Then X generates G if and only if there is no maximal subgroup of G containing X. The set X is a minimal generating set of G if and only if its image in V forms a basis for V. Thus all minimal generating sets of G have the same size, namely dimFp V. In particular, this applies to p-groups and, more generally, pro-p groups.
The basic idea is to capture a large part of the group structure in a Lie ring. Standard examples of Lie algebras include matrix algebras. However, one should think of a Lie algebra essentially as a vector space.
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We trust that the reader will make the appropriate translations of this kind where necessary; e. If all the homogeneous components Li happen to have exponent p, we can regard L even as a Lie algebra over Fp.
An example of this construction is described in Exercise 4. Compact p-adic Lie groups denote the usual formal power series. Without specifying further details at this point, we formulate: Theorem 2. The correspondence preserves such invariants as the orders and the nilpotency classes of the objects involved. We claim that in addition to these there are up to isomorphism precisely two non-abelian groups of order p3.
Clearly, the underlying additive group of such a Lie ring L cannot be cyclic. From this one shows that each of the two non-cyclic abelian groups of order p3 supports essentially one nilpotent Lie ring structure.
In particular, 2. Note that the graded Lie rings associated to G4 and G5 with respect to their lower central series coincide. It turns out that they do in fact determine the pro-p groups completely; see Section 8.
Intuitively, x is p-adically small if it is divisible by a large power of p. A theorem of Ostrowski states that, up to a suitable equivalence, the ordinary absolute value and the p-adic absolute values exhaust all possible non-trivial 16 Chapter I. We remark that base 10 is chosen by convention, not for any intrinsic mathematical reason.
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In this case, one has to adjoin all the missing limits of equivalence classes of Cauchy sequences with respect to dp. Similarly, one extends to Qp the valuation map vp and the ring operations, addition and multiplication. The elements of Qp are called p-adic numbers. A convenient notation for explicit computations with p-adic numbers is the following. Section 3 contains a summary of basic notions in topology, which we assume. Approximate computations in R can be performed by truncating the decimal representations of the numbers involved and verifying that errors do not pile up too much — the last bit can actually be quite tricky.
Indeed, bounding error terms is one of the main themes in analysis. In a similar way, the p-adic absolute value induces a metric and hence a topology on Qp. It is the topological closure of the ordinary integers Z in Qp , and its elements are called p-adic integers. The structure of the ring of p-adic integers is quite simple. In Section 5. Nevertheless, skipping the theory that lies in between, we can already formulate a precise and hands-on description of the family of compact p-adic Lie groups and p-adic analytic pro-p groups, in particular. A topological group is a group G which is also a topological space such that the group operations are continuous, i.
Matrix multiplication is easily seen to be continuous, and so is the process of forming the inverse of an invertible matrix. Hence GLd Zp , equipped with the subspace topology, becomes a topological group. We can now state: Theorem 2. A compact topological group admits a p-adic analytic structure if and only if it is isomorphic to a closed subgroup of GLd Zp for a suitable degree d. In fact, in his seminal paper  Lazard established a whole theory of p-adic analytic groups with much wider consequences.
One of his key results is that the analytic structure of a p-adic analytic group is determined entirely by its topological group structure. As we will see, GLd Zp is virtually a pro-p group. Theorem 2. We conclude this section with a concrete reformulation of Theorem 2. Section 5.
One particular Sylow p-subgroup of GLd Fp is the group of 3. Notions and facts from point-set topology 19 upper uni-triangular matrices; according to the Sylow theorems, all other Sylow p-subgroups of GLd Fp are conjugate to this one. Corollary 2.
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A p-adic analytic pro-p group is a topological group which is isomorphic to a closed subgroup of a Sylow pro-p subgroup of GLd Zp for a suitable degree d. To discuss their structure, we require basic notions and facts from point-set topology. For the convenience of the reader, I have listed the relevant prerequisites below. Suitable references for general topology are, for instance, [5, 28, 59].