A group which does not belong to but all of whose proper quotient groups belong to is called justnongroup. A profinite group is called small if it has only finitely many open subgroups of. In this context, an inverse system consists of a directed set. We consider profinite groups as 2sorted firstorder structures, with a group sort, and a second sort that acts as an index set for a uniformly defin. We then consider other generationrelated profinite properties such as lower rank and finite presentability. In the case of a finite group g, we prove that if, for some n, jengj 6 m for all g 2 g, then the order of the nilpotent residual 1g is bounded in terms of m. Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We also give a quick survey of some areas of infinite group theory in which these and related constructions based on finite simple groups play a role. We develop properties of goodness and establish goodness for certain important groups. We treat g k as a fundamental object of study because it allows us to control all separable extensions l of k in one stroke. It is proved that all subgroups of finite index in a profinite group g are. Its one of several programs that uses the highly structured ortongillingham approach. On profinite groups with engellike conditions sciencedirect. Wilson, profinite groups oxford university press, oxford, 1998.
Strong completeness of profinite groups was studied by several authors, including pletch in 2, ribes and zalesskii in 3, and wilson in 4. This is the first book to be dedicated solely to the study of general profinite groups. Maximal subgroups and strong completeness of profinite groups. As an unexpected byproduct our groups yield new examples related to a question of grothendieck concerning isomorphism of groups with isomorphic profinite completions.
Finite axiomatization of finite soluble groups wilson, j journal of the london mathematical society issue 3 volume 74 page 566582 december 2006 research interests. Wilson, profinite groups, clarendon press, oxford 1998. Recall that a group is periodic if all of its elements have finite order. A profinite group g is just infinite if every nontrivial closed normal subgroup of g is of finite index, and hereditarily just infinite if every open subgroup. Buy profinite groups london mathematical society monographs 19 on. An exposition of the connection between limitperiodic. Almost engel finite and profinite groups the lincoln. In mathematics, profinite groups are topological groups that are in a certain sense assembled. Profinite groups london mathematical society monographs book 19 kindle edition by wilson, john s download it once and read it on your kindle device, pc, phones or tablets. Profinite groups london mathematical society monographs 19. The positive solution of the restricted burnside problem, had led to many remarkable results on profinite groups. The underlying theory reflects these diverse influences, with methods drawn from both algebra and topology and with fascinating connections to field theory.
In particular, using wilsons reduction theorem, zelmanov has been able to prove local finiteness of profinite periodic groups. For prop groups a convenient source is the book of dixon, du sautoy, mann and segal ddms. Commutators and pronilpotent subgroups in profinite groups. Elementary equivalence of profinite groups by moshe jarden. Moreover, we show that the direct product of certain hereditarily just infinite groups of finite lower rank still has finite lower rank. Profinite groups with nip theory and padic analytic. Finite index subgroups and verbal subgroups in profinite groups by john s. Wrs is designed for students from second grade up to adulthood. This is the second workshop in the winter of disconnectedness programme, a series of workshops and seminars on the topic of totally disconnected, locally compact t. A profinite group is called small if it has only finitely many open subgroups of index n. We find explicit generators for this family in a number of cases using combinatorial methods. As such they are of interest in algebraic number theory. Galois groups carry with them a natural topology, the krull.
Wilson, on subgroups of finite index in compact hausdorff. Geom dedicata profinite rigidity for seifert fibre spaces gareth wilkes 0 0 mathematical institute, university of oxford, andrew wiles building, radcliffe observatory quarter, woodstock road, oxford ox2 6gg, uk an interesting question is whether two 3manifolds can be distinguished by computing and comparing their collections of finite covers. Finally we examine whether the groups in this family admit a finite profinite presentation. Finiteness properties of profinite groups 2010 cached. Get your kindle here, or download a free kindle reading app. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further. Request pdf finite index subgroups in profinite groups we prove that every subgroup of finite index in a topologically finitely generated profinite group is. For background on profinite groups we recommend wilsons book wi and the book of ribes and zalesskii rz, where detailed proofs and generalizations can be found for the results stated below. Wilson 15 made a study of just infinite groups possessing no nontrivial abelian subnormal subgroups and returned to it in a recent paper 16 where he also considered just infinite profinite. The wilson reading system wrs is an instructional program used to help struggling readers. Next we turn our attention to groups of unipotent automorphisms of residuallyfinite and profinite groups. We study a family of hereditarily just infinite profinite groups obtained by iterated wreath products introduced by j. Please submit your resume using the form below and our hiring manager will contact you if.
Citeseerx finiteness properties of profinite groups. Commutators and pronilpotent subgroups in profinite groups commutators and pronilpotent subgroups in profinite groups detomi, eloisa. If the inline pdf is not rendering correctly, you can download the pdf file here. Profinite groups can be defined in either of two equivalent ways. But its most widely used with upper elementary students, middleschoolers and highschoolers. This property relates the cohomology groups of a group to those of its profinite completion. As a subspace of the direct product, endowed with the compact topology a neighbourhood base of the identity is given by the system of kernels of the projections, it is closed and hence compact. On groups whose subgroups are closed in the profinite topology on groups whose subgroups are closed in the profinite topology robinson, derek j. We show that two profinite groups that are elementarily equivalent, in. Wilson, on subgroups of finite index in compact hausdorff groups. First, this class, being defined by restricting the length of certain normal series, contains several natural classes of profinite groups like the class. Modular representations of profinite groups sciencedirect. Torsion in profinite completions connecting repositories.
Indeed, profinite groups are precisely galois groups and many of the applications of profinite groups are related to number theory. Thanks for your interest in a career with the wilson group. The definition of the class of groups of fixed abeliansimple length appears in 2, section 2, and is given in the section devoted to these groups in our work section 5, definition 5. An infinite group g is called just infinite if all nontrivial normal subgroups have finite index. Profinite groups are of interest to mathematicians working in a variety of areas, including number theory, abstract groups, and analysis. One of the consequences of the solution of tarskis problem is that all. The present note is a survey of recent results on the topic with a special attention to topological groups.
Wilson, profinite groups, clarendon press, oxford, 1998. This website is best viewed in 800 x 600 resolution. Profinite graphs and groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. We show that such groups are locallynilpotent and, as a byproduct, we obtain an alternative proof of a well known theorem of wilson about nengel residuallyfinite groups.
Nilautomorphisms of groups with residual properties. A survey on justnon groups hindawi publishing corporation. Moreover, the groups q and q can be identified with ga and gaa, respec tively. Profinite groups with an automorphism whose fixed points are right. We are thus interested in profinite modules over the completed group algebra k g of a profinite group g, where k is a finite field of characteristic p. Wilson, profinite groups, london mathematical society monographs. Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. On just infinite abstract and profinite groups request pdf. We then discuss determination of the lower rank for this family. New series 19 the clarendon press, oxford university press, new york, 1998. A topological group that is the projective limit of an inverse system of finite discrete groups, where is a preordered directed set.
Profinite groups with nip theory and padic analytic groups dugald macpherson. Use features like bookmarks, note taking and highlighting while reading profinite groups. The reason to consider this class of groups is twofold. Profinite groups london mathematical society monographs. We show that some generalised wilson groups are new examples of profinite groups with finite lower rank. On just infinite abstract and profinite groups springerlink. We also show that any nip profinite group, presented as a 2sorted structure, has an open prosoluble normal subgroup. On generation problems in generalised wilson type groups. Groups, combinatorics and geometry world scientific. Torsion in profinite completions 147 subgroup of g. On hereditarily just infinite profinite groups that are.
A profinite group is an inverse limit of a system of finite groups the finite groups are considered as compact discrete topological groups and so the inverse limit, as a closed subspace of the compact space that is the product of all those finite groups has the inverse limit topology, hence is, as is said above, a compact hausdorff, totally disconnected group. We prove that this family contains groups of finite lower rank. Free subgroups of finitely generated free profinite groups. Profinite rigidity for seifert fibre spaces pdf paperity. A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.
We show that two profinite groups that are elementarily equivalent, in the. A course in model theory by katrin tent cambridge core. Finite index subgroups in profinite groups request pdf. We also show that many groups in this family are not topologically finitely presentable. On groups whose subgroups are closed in the profinite topology. Use features like bookmarks, note taking and highlighting while reading profinite groups london mathematical society monographs book 19. The proof uses the wilsonzelmanov theorem saying that engel profinite groups are locally nilpotent.
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