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FREE PRO-C-GROUPS by Lim Chong-Keang Ph.D. Thesis In

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FREE PRO-C-GROUPS by Lim Chong-Keang Ph.D. Thesis In 1· FREE PRO-C-GROUPS by Lim Chong-Keang Ph.D. Thesis in Mathematics, McGill University, Montreal, Canada. 1971 ABSTRACT For any class C of finite groups, closed under the formation of subgroups, quotients and finite products, we .1 consider the category PC of pro-f-groups, i.e. inverse limits of f-groups. We construct left adjoints F to forget~ functors from PC to various underlying categories and prove tripleableness ., , theorems for these adjoint pairs. A necessary and sufficient ~, .; ", condition for a topological space X to be embedded in F(X) is found. A formula for the cardinality of the free prof1nite group on any set is obtained. We look at certain closed subgroups of F(X) , in particular, we prove that the Sylow p-groups G of a free profinite p group, generated by a set, are free pro-p-groups in the sense of Serre, and we obtain a formula for the minimal number of generators of G • A characterization of pro-nilpotent groups p 1s given and it is shawn that the free pro-nilpotent group on n generators is isomorphic to the closedmult1plicative subgroup of î{{T ,T , ••• ,T }} generated by 1+T ,1+T , ••• ,1+T • l 2 n l 2 n :f '" " SHORT TITLE FREE PRO-C-GROUPS Lim Chang-Keang l 1 1 1{ } \ J J 1 nEE PRo-C-GROUPS by LimChong-Keang A Theais Submitted in Partial Pulfilment of the Requirements for the Degree of DOCTOR OP PHlLOSOPHY Department of Kathematies HcGill University Montreal, Quebec, canada. , 197 1 (ii) 1 TABLE OF CONTENTS INTRODUCTION NOTATIONS CHAPTER 0 PRELIMINARIES ·............ 1 CHAPTER 1 PRO-C-GROUPS ·............ 8 CHAPTER 2 PREE PRO-C-GROUPS ·............ 18 l CHAPTER 3 SUBGROUPS OF PREE PRO-f-GROUPS ....... 46 1 CHAPTER 4 PRO-NILPOTENT GROUPS 63 ·............ 1l 1 1 CHAPTER 5 TRIPLEABLENESS ·............ 77 l l 1 REFERENCES ·............ 94 1 1 \ 1 l j j 1 1 ! 1 1 1 (Hi) INTRODUCTION In a Galois extension (possibly infini te) E/K, it is known that the Galois group G(E/K) of E over K is isomorphic to the inverse limit of the inverse family of finite groups G(Fi/K) where Fi ranges over aIl subfields of E containing K such that Fi/K is a finite Galois extension. If E/K is a finite Galois extension, then there is a one-one correspondence between the subfields of E containing K and the subgroups of the Galois group G(E/K). However, this correspondence breaks down in the case of infinite Galois extensions. In order to overcome this difficulty, Krull introduced a topology on G(E/K) such that there is a one-one correspondence between the subfields of E containing K and the closed subgroups of G(E/K). In this case, G(E/K) turns Dut to be a compact, Hausdorff, tota1ly disconnected topological group, and hence a profinite group. The cohomology theory of profinite groups has been studied extensively in Douady [11, Lang [11 and Serre [11. In 1969, Binz-Neukirch-Wenzel [11 extended the definition of profinite groups to a wider c1ass, pro~-groups, which includes in particular profinite groups, pro-abelian groups, pro-p-groups, pro-s01vable groups, pro-nilpotent groups, etc. Uv) The purpose of this thesis is to study the structure of free pro-f-groups. By a pro-f-group, we mean a profinite group G such that GIN € Ifl for all open normal subgroup N of G, where C is a full subcategory of the categroy of finite discrete groups and continuous homomorphisms. Throughout this thesis, we shall assume that f is closed under subobjects, quotient objects and finite products. With these restrictions on f, the category PC of pro-f-groups and continuous homomorphisms is shawn,in Chapter l, to be closed under subobjects, quotient objects and products, with respect to the category PF of profinite groups and continuous homomorphisms, and hence constitutes an example of a variety in PF. ln Chapter 2, we construct left adjoints to forgetful functors from PC to various underlying categories. We prove that, for a compact space X, the necessary and sufficient condition for X to be embedded in the free pro-f-group generated by X is that X should be Hausdorff and totally disconnected. A formula for the cardinality of a free profinite group generated by an arbitrary set is obtained. In Chapter 3, we look at certain closed subgroups of free pro~-groups. ln particular. we prove that the Sylow p-groups G of free profinite groups generated p by a set are free pro-p-groups in the sense of Serre [1], and 1 (v) we ob tain a formula for the minimal number of generators of G • P In Chapter 4, we give a cbaracterization of pro-nilpotent groups, and we prove that the free pro-nilpotent group on n generators is isomorphic to the closed multiplicative subgroup of the non-commutative ring of formaI power series T ,T ,····,T }} generated by l+T ,l+T , ••••• ,l+T • t {{ l 2 n l 2 n The last Chapter is devoted to study the tripleableness of the forgetful functors from PC to various underlying categories. Finally it is shown that PC is equivalent to the category of algebras of the theory of the forgetful functor from f to! (the category of sets and mappings). This last result was proved by Professor Dion Gi1denhuys and conjectured by Prof essor M. Barr for the case f = f, the category of finite discrete groups and continuous homomorphisms. The author expresses his heartful thank to his supervisor, Prof essor Dion Gi1denhuys, for his valuable guidance and patience in the preparation of this thesis, without which this thesis could not have been written. Gratitude is a1so extended to the National Research Çouncil of Canada for financia1 assistance in the form of a three-year postgraduate scholarship, to my wife Din-Din and members of my family, for their constant encouragement. 1 (vi) NOTATIONS 5 Category of sets and mappings. G Category of groups and homomorphisms. TG Category of topo10gica1 groups and continuous homomorphisms. Category of topo10gica1 spaces and continuous mappings. Category of pointed topo10gica1 spaces and continuous mappings preserving base point. Category of compact Hausdorff topo10gica1 spaces and continuous mappings. Category of comp1ete1y regu1ar T -spaces and continuous 1 mappings. Fil Category of sets with fi1ters whose objects are ordered pairs (5 ,M), where 5 is a set and Ma filter on 5. A morphism f:(5 ,M ) ~ (5 ,M ) is a map f:5 ~ 52 such that 1 1 2 2 1 f-lv €M for a11 V € M • 1 2 F Full subcategory of TG whose objects are finite discrete groups. C Full subcategory of F c10sed under subobjects, quotient objects, and finite products. PF Category of profinite groups and continuous homomorphisms. PC category of pro~groups and continuous homomorphisms. Objects of the category !. Set of aIl morphisms fram X to Y in A. (vii) r F --1 U F is the lef t adjoint of U. Z Set of aU integers. N Set of al! natural numbers. i The p-adic integers, lim z/p z • ~ -+- -- 1\ Z 1!tD ~-'nz n€Z A Closure of A. flx Function f restricted to X. Ker f Kernel of the function f. lm f Image of the function f. ::. Isomorphic. U Disjoint union. GIH Factor group. X\A Set difference. L(X) Free discrete group on X. FC(X) Free pro~-group on X. Ixi Cardinali ty of X. >to Aleph null, the smallest infinite cardinal. iff if and only if. l.c.m. least common multiple. Equivalence between categories. -1- CHAPTER 0 PRELIMINARIES ln this chapter, we gather some basic resu1ts, that will be used in the seque1. The detai1s can be found in one or more of the fo1lowing: Bourbaki [1], Montgomery & Zippin [1], Dugundji [1], or Bucur & Deleanu [1]. Definition 0.1. Let A be an arbitrary category, and 1 a small category. The inverse limit 11m D of a functor D : 1 ~ ! is an object L of!, together with a natural transformation t from the constant functor L at L to D, such A that if also L' € I!I and t' : L' ~ D is a natural trans- formation, then there exists a unique morphism Q : L' ~ L in A such that t(i)oQ • t'Ci) for aIl i € Ill. ( The constant functor L maps all objects of 1 onto L, and it maps al1 morphisms of lonto the identity map of L ). One easily sees that 1 l!m : A- ~ A is the right adjoint of the functor 1\ : -2- where the functor A maps every object A in ! onto A, and ,.. maps every morphism a in A onto a, a(I) = a for all I ! I!I. The inverse limit of a functor is obviously unique up to isomorphisme Definition 0.2. A poset is an ordered pair (A,S) where S is reflexive, anti-symmetric and transitive. A poset (A,S) is said to be directed if for any a,b in A there exists c in A with aSc and bSc. Remark 0.3. If I is a poset, then I can be looked upon as a (small) category !' whose objects are the elements of land such that !(i,j) is empty if i is not Sj, and !(i,j) con tains exactly one element if iSj. Definition 0.4. A directed inverse system in ! is a contravariant functor from a directed poset ! to!; i.e. it is a functor from !op to !.
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