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)

number of generators of G • we ob tain a formula for the minimal 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 }} generated by l+T ,l+T , ••••• ,l+T • t {{ Tl ,T2 n l 2 n The last Chapter is devoted to study the tripleableness

of the forgetful functors from PC to various underlying the categories. Finally it is shown that PC is equivalent to

category of algebras of the theory of the forgetful functor result from f to! (the category of sets and mappings). This last

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 a Çouncil of Canada for financia1 assistance in the form of

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 or will be used in the seque1. The detai1s can be found in one 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 natural D : 1 ~ ! is an object L of!, together with a transformation t from the constant functor L at L to D, such

A trans- that if also L' € I!I and t' : L' ~ D is a natural

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 !. Denoting the value of the functor

at i ! I!I by Ai' one bas the following equivalent description

of a directed inverse system as a famui { Ai' fj,i' j~i in Il satisfying the following properties:

1 -3-

(a) (I,S) is a directed poset,

(b) . {Ai : i ~ I} is a family of objects in !, (c) For each pair of indices i,j in l with i Sj,

fj,i : Aj -+ Ai is a morphism in !, and

fj,ifk,j .. fk,i

whenever iSjSk •

(d) For each i ~ l, fi,i is the identity morphism on Ai •

Sometimes we write' {Ai : i ~ I} instead of

, {Ai' fj,i' j~i in I} when it is clear what the maps are.

Definition 0.5. A directed inverse limit in A ls

the inverse limit of a directed inverse system in!. If

. {Ai : i € I} is such a directed inverse system in!, we shall

denote the directed inverse limit by l1m' { Ai: i € I} or

simply l!m Ai •

Proposition 0.6. Let! be one of the categories Top or TG, let DIoP -+ A be a functor, and l a small

ca tegory. Then

l1m D =' { a € fi D(i) D(f)(Pj(a» - Pi(a),whenever f:j-+il , i~I!1 -4-

where the Pi : n D(i) + D(i) (i € Ill) are the canonical i€III projections, and the topology on l!m D is induced by the product topology on the product n D(i). i€III

Proposition 0.7. Let' {Ai : id} be a directed inverse system in Top and J a cofinal subset of 1; let

Pj: l!m Ai + Aj be the canonical mapping. Then

(a) l!m' {Ai : i € I} == lim' {Aj : j € J} -1 (b) the family of sets Pj Uj , where j runs through

J and Uj runs through a base of the topology of Aj (j€J), is a base of the topology of lim Ai •

A topological space X is cOmpact if every open cover of X admits a fiuite subcover. ( X need not be Hausdorff.)

A topological space X is totally disconnected if the

connected component of each point of X consists of the point

alone.

Proposition 0.8. Let G be a topological group with

identity element e. -5-

(a) G is Hausdorff if and only if the intersection of the neighbourhoods of e is' { e}.

(b) G is totally disconnected if and only if the connected component of e is' { e}.

(c) If H is a normal subgroup of G, then the quotient

group G/H is Hausdorff (respectively discrete) if and only if H is closed (respectively open) in G.

(d) If' {U : id} is a fundamental system of i neighbourhoods of e, then for any subset S of G,

S. n SU a id i

(e) If A 18 closed and K a compact subset of G, then

KA 1s closed in G. (f) If G 1s compact, Hausdorff and totally

d1sconnected, then the family of all open normal subgroups of G

constitutes a fundamental system of ne1ghbourhoods of e.

(g) If G is compact, Hausdorff and totally

disconnected, then so are its closed subgroups, its quotients

by closed normal subgroups.

s -6-

Proposition 0.9. Let! be a small category,

D : 10P ~ Top a functor and X = llm D. For each i€I!I(-I), write Xi - D(i). (a) If each Xi is Hausdorff, then X is closed

(b) If each Xi is compact, Hausdorff, then so is X. (c) If 1 is a directed poset and each Xi is compact, Hausdorff , then X is non-empty and

where Pi X ~ Xi is the canoniesl projection.

Proposition 0.10. Assume that'{Ai , fj,i' ja in Il is a directed inverse system in Top, A - I1m Ai ' and

Pi: A ~ Ai denote the canonical projection (i€I). Suppose

B€ITopl and Qi : B ~ Ai are morphisms defined for each i€I, and

such that fj,iaj - ai whenever iSj in Ij and suppose, furthermore,

that each ai i8 surjective. Then the map a : B ~ A, induced by

the maps ai ' has dense image in A. 50, if B is compact and A is Hausdorff, then a is surjective.

Proposition 0.11. Let X be compact and Y be Hausdorff

in Top • then any bijective continuous map f : X .. Y is a

homeomorphi81D • -7-

Proposition 0.12. Let G be a compact, Hausdorff and tota11y disconnected topological group. For every pair

N, N'of open normal subgroups of G, with NcN', one has a canonical homomorphism GIN ~ GIN' of discrete finite groups, and the canonical projections G ~ GIN induce an isomorphism

G = l!m GIN of topological groups, so that G is a directed inverse limit of finite groups; i.e. G is profinite.

Conversely, every profinite group is compact,

Hausdorff and totally disconnected. -8- r.

CHAPTER 1

PRO-C-GROUPS

Let F be the full subcategory of TG whose objects

are finite discrete groups, and let f be a full subcategory of I, which is c10sed under subobjects, quotient objects and finite products. Objects of f will be ca11ed f-groups.

Examp1es of f are (i) a11 finite groups (ii) finite abe1ian

groups (iii) finite p-groups (iv) finite solvable groups

(v) finite nilpotent groups, etc.

Proposition 1.1. Let G be a topo1ogiea1 group. The

fo11owing are equiva1ent:

(a) G is compact, Hausdorff and admits a fami1y ~

of open normal subgroups of G, such that ~ is a fundamenta1

system of neighbourhoods of the identity, and has the property

that for each NE~, GIN E Ifl •

(b) G is profinite and GIN Elfl for every open normal

subgroup N of G. (c) There exists a sma11 category l and a functor

DI'" C such that G ~ 1i:2D D in TG • + - (d) G is a directed iDverse 1tmit (taken in TG)

of groups in C•

• ,-, .. -·· ..·-· .. • ...... ___ 1 ,. __.. , ______.~ ... -9-

~: (a) ~ (~l Let~' consist of aIl finite

intersections of meabers of ~. The map

induced by the canoniesl projections G ~ GIN, N€~', is surjective,

by Proposition 0.10, and its kernel is

n N • n N • the identity Nd' Nd

by Proposition 0.8(a). Thus ~ is an isomorphism in TG. The family

~, is directed by inclusion since it is closed under the

formation of fiuite intersections. Given Ni ,Ni , •• ,Ni €~, 1 2 n we note that GI RNi is isomorphic to a subgroup of j-l j

jnlG/Ni ,hence lies in I~I. Thus GIN €I~I for each N€~' and j

hence G is isomorphic to a directed inverse limit of groups in Ifl.

(d) ~ (c) is trivial.

(c) ~ (a) By Proposition 0.9, G is compact

and Hausdorff. If V 1s an open neighbourhood of the identity,

then there ex1st 1l ,12, •••• ,in €Ill such that Pi(V) • D(i) for aIl i~il'i2' •••• ,1n; where Pi : G ~ D(i) are the canonical

projections. If we denote the identity element of D(ij ) by ej , then

c V

and N i8 an open normal subgroup of G. Horeover, the projections r- -10-

Pi induce an embedding j GIN

Since f is c10sed under the formation of finite products and subobjects, we have GIN €Ifl, and this proves that the fami1y

~ of a11 open normal subgroups N of G, having the property that GIN €Ifl, constitutes a fundamenta1 system of neighbourhoods oi the identity.

(a) ~ (b) Since every open subgroup is c10sed, it fo11ows that each N€~ is c10sed in G. Thus, the fami1y of a11 open and c10sed neighbourhoods of the identity e has intersection , { e }, and so G is totally disconnected. By hypothesis (a),

G is compact and Hausdorff so that it is profinite, by

Proposition 0.12. If V is any open normal subgroup of G, then

V contains a member N of t. Since G/V is a quotient of G/N€lfl, we have G/V € Ifl. This proves (b).

(b) ~ (a) By Proposition 0.12, G is compact, Hausdorff and tota11y disconnected. Take for t the fami1y of a11 open

normal subgroups of G. Then the resu1t fo11ows fram Proposition O.S(f).

Definition 1.2. A topologica1 group G, satisfyinl the

equiva1ent conditions of Proposition 1.1, will be ca11ed a

pr~-Iroup. -11- - [

The category PC is defined to have as its objects the

pro~-groups, and as its morphisms all continuous homomorphisme,

so that one has the following full and faithful embeddings:

C ~ PC ~ TG

If C consists of all finite p-groups, the pro~-groups are

called pro-p-groups.

Proposition 1.3.

(a) A subgroup of a pro~-group w1th induced topology 1s a pro-f-group if and only 1f it is closed.

(b) The quotient of a pro~-group by a closed

normal subgroup 1s a pro~-group.

(c)A product of pro-C-groups is a pro~-group. (d) For every small category l, and functor

D 1 ~ PC, the inverse limit of D exists in PC.

Proof: (a) Let H be a closed subgroup of a pro~-group

G. Then by Proposition 0.8(g) and Proposition 0.12, G is profinite. If N is an open normal subgroup of H, then N-onH, for some open neighbourhood 0 of the identity. By

Proposition 0.8(f) and Proposition 0.12, there exists an open

normal subgroup U of G, such that UoD. Therefore, UnH c OnH - N, and so the group H/UnH is a subobject of GlU. -12-

r'" ,

Since GlU € le l, therefore H/UnH be10ngs to Ifl, and so does

its quotient B/N. Thus H satisfies Proposition 1.1(b), and

hence H is a pro~-group. If H is a subgroup of Gand is a

pro~-group, then it must be c10sed, since it i8 compact.

(b) Let G be a pro~-group and let H be a c10sed

normal subgroup of G. By Proposition 0.8(g), G/H is profinite.

If N is an open normal subgroup of G/H, then N bas the form u/H for some open normal subgroup U of G. This imp1ies that

(G/H)/N - (G/HV(U/H) ~ GlU , and so (G/H)/N belongs to Ifl,which proves that G/H satisfies

Proposition 1.l(b), hence the resu1t fo11ows.

(c) Let G - TI G be a product of pro-f-groups, i id

and let N be an open normal subgroup of G. !ben there exist

.;.{ Ni for al1 i-il'i2, ••• ,in• PiN G otherw1se. i where Ni are open normal subgroups in Gi and Pi: G ~ Gi are the canonical projections. The composite maps

G ~ Gi + Gi/PiN induce an injection

GIN - -13-

Since f is c10sed under the formation of finite products and subobjects, we have GIN e: Ifl, and this proves that G i8 a pro-f.-group.

(d) By Proposition 0.9(a), the inverse 1imit 11m 0 is a c10sed subgroup of n D(i), and so the resu1t fo11ows ie: III from (a) and (c).

Proposition 1.4.

(a) In PC, "one-one" i8 equiva1ent to "mono".

(b) In PP, "onto" is equiva1ent ta "epi".

Proof: (à) The fact that "one-one" implies "mono" is we11 known and the proof is straightforward. Por the converse, let f: G'~ G be mono in PC. Let K-Ker f, then K is a closed normal subgroup of G'. By Proposition 1.3(a), K is a pro-f.-group. Define a,a : K ~ G' to be the inclusion map and the trivial map of K into G' respectively, then a,S are morphisms in PC if we consider K as a subspace of G'.

Thus we have fa - fa in PC, and hence a - B since f 1s mono.

Hence K· ~} , i.e. f is one-one. (b) "ODto" implies "ep1" is straightforward.

For the converse, let f: G ~ G' be epi in PP. For any open normal subgroup N of G', G' IN and G/f-~ both belong to Ifl. -14-

We first show that the map

f : G/f-~ ~ G'/N N induced by f is epi in F. To prove this, let H€I!I and let a,S : G'/N ~ H be morphisms in! with the property that afN - SfN· One has the following diagram:

f N a GIC~ G' IN H - -~

1T N 1T N'

G - G' f qbere 1T , 1T ' are canonical quotient maps. So we have N N

a1T N'f - afN1TN - SfNlI'N • S1TN'f which imp1ies that a1T N' - SlI" N since f is epi. The map 1T N' is surjective so that a-S and this proves that fN is epi in F. However, it is weIl known that in!, "epi" implies

"onto" (see e.g. Mitchell [1» so that f N is onto. Thus, the composite map 11' f G ~N GIC~ ~N G' IN

is oU!O, for every open DOrmal subgroup N of G'. By Proposition 0.12 -15-

(

there is an isomorphism G' ~ l!m G'/N making the following

diagram commute:

f :: G G'

oN 1

G/f-~ ~ G' IN f N where PN is canonical, and it now follows from Proposition 0.10 that f is onto. The proof is now complete.

Corollary • PC as a full subca tegory of PP is closed under subobjects, quotient objects and products. Hence is a variety and a full reflective subcategory of PP, by Birkhoff's theorem.

The following Proposition describes the reflector R of --PP in PC •

Proposition 1.5. The inclusion functor PC ~ PP

has a Ieft adjoint R : PP ~ PC, described as follows: For

every profinite group G, denote by t the family of those

( ; -16-

-L

open normal subgroups N of G for which GIN €Icl. Then R(G)· 1!m GIN N€~

Proof: We need oo1y verify that the canonica1 map

n : G + R(G) has the universal property of a front adjunction. G

So let H bea pro-f-group and f : G + H a continuous homomorphisme For every open normal subgroup N of H, f induces

an injective homomorphism

f : + HIN N G/f-~ hence f-~ €~. Let

be the canonica1 projection. The maps fNPN ' where N ranges over a11 open normal subgroups of H, induce a map

~: R(G) + l!m HIN

80 that on.e has a cOlllDUtative diagram:

f G 1 H

nG "\"N

1l(G) .11- HIN

PHi [PH' C f N GIC~ 1 HIN

"-- -_···.:.. 7· - --." -17-

,... 1

the maps PN' and n being canonica1. Thus, there exists a N

morphism ~: R(G) ~ H such that ~nG = f. By Proposition 0.10,

n is surjective, so that ~ is the unique morphism for which G

~nG a f. This comple~es the proof of ~he Proposition. -18-

CRAnER 2

FREE PRO-~-GROUPS

Proposition 2.1. Let U : PC ~ PTop be the forgetful functor which maps a pro-C-group G onto (G,e) where e 1s the identity element of G. Then U has a left adjoint topological F : PTop ~ PC described as follows: For any pointed space (X,x), let L be the free discrete group with X\~} as basis and x as its identity. Denote by .X the family of al1 normal subgroups N of L for which LIN €Ifl and such that all aNnX, a€L are open. Then F(X,x) - l!m LIN N€41X

Proof: By proposition 1.l(c), one has F(X,x)€IPC\.

There is a map (X,x)

induced by the composite maps 1fN (X,x) l. (L,x) ~ U(L/N)

L,obta1ned where \ i8 the extension of tbe canonical embedding X\ bel ...

1f (N€t ) are by sending x to the identity e1ement of L, and N X canonical. The condition that all &NnX, a€L are open implie8 U, that 'lX 18 continuous. To show that F 18 the left adjoint of -19- r

we need to verify tbat nx bas the univers al property of a

front adjunction. Let G be a pro-C-group and f: (X,x) ~ (G,e)

a morphism in PTop • Then f induces a homomorphism r: L ~ G.

Denote by ~ the family of aIl open normal subgroups of G. One

bas injective continuous homomorphisms f : GIN N L/r-~ ~ (N€~) induced by r. Since(a.r-~)nx = f-l(l'(a).N), it follows that (a.l'-~)nx is open for aIl a€L, hence l'-~ € ~X. Let PN : F(X,x) ~ L/l'-~ (N€t) be the canonical projections. The composite maps fNPN (N€t) then induce a continuous homomorphism

~ : F(X,x) ~ 11m GIN so that one has a commutative diagram: Nd

(X,x) f (G,e) ~j l' UF(X,x) u~!~ 1 U(Um GIN) U('II ) .. N N€~

U(PN)! lU(PN ')

U(f ) U(L/r-~) B U(G/N)

, Thus, there exista a where the maps PN ' 'liN are C8DOnical. -20-

cont1nuous homo1DOrph1sm '" : F(X,x) -+ G su ch that "'nx .. f. It remains to show that '" is unique with this property. Suppose

",' : F(X,x) -+ G is a continuous homomorphism with ""nx = f. Then we have two h01DOmorphisms L ! F(X,x) t G R m' L ~ F(X,x) + G whose restrictions to X\ fx} are both equa1 to f IX\ {x}, and a is induced by the canonica1 projections L -+ LIN (Ne$x). Thus by the freeness of L, one has ",a .. ",'a and hence '" - ",' on the image of L in F(X,x) under a. From Proposition 0.10, aL is dense in F(X,x) and so '" • ",', since

G is Hausdorff. The proof is DOW complete.

The forgetful functor PTop -+ Top has a 1eft adjoint which sends every topo10gical space X onto (Xu{*l,*), where

Xu {*} is the coproduct in Top of X and a single point *.

The composition of two adjoint pairs, when defined, yie1ds another adjoint pair, so one bas the fo11owing:

Proposition 2.2. The forgetful functor PC -+ Top has a left adjoint F : Top -+ PC. For any topologica1 space X

F(X) - 1im.- LIN Netx P \ -21-

where L is the free discrete group with X as basis, and ~X

denotes the fami1y of a11 normal subgroups N of L such that

(i) LIN €Ifl and (ii) aNnX, a€L are a11 open.

Coro11ary. The forgetfu1 functor PC ~ S has

a 1eft adjoint F: S ~ PC. For any set X,

F(X) = 1!Jn LIN

where L is the free discrete group on X and N ranges over a11

normal subgroups of L such that LIN €Ifl.

There is a functor E: PTop ~ Fil which sends

every pointed topo10gica1 space (X,x) to (X,M) where M is the

fi1ter of neighbourhoods of X; the action of E on morphisms is defined in the obvious way. Then E bas a 1eft adjoint

T : Fil ~ PTop defined by T(S ,M) - (Su {*}, *), where * is

an e1ement not in S; the topo10gy on Su{*} is defined as

fo11ows: (i) every point in S is open (ii) V is an open

neighbourhood of * if and on1y if * €V and V\ {*}€M • From

Proposition 2.1, one then bas the fo11owing:

Proposition 2.3. The composite functor U E PC -+ PTop -+ Fil -22-

(

Por any (S,M) €IFill, has a 1eft adjoint P : Pi1 ~ ~.

lim LIN F(S,M) • + Nd S

on S, and ~s denotes the where L is the free discrete group N of L such that family of those normal subgroups Nns , be10ng to M. (i) LIN € Ici and (ii) a11

group, let Definition 2.4. Por any topo10gical 1\ G - l!m GIN

normal subgroups of G with wbere N runs through a11 open and is called the GIN € Ifl. Then e is a pro-f-group

pro-~completion of G.

~ G induced by There is a canonica1 map n : G " N being the open normal subgroup ;\ the quotient maps G ~ GIN , a genera1ization of of G with GIN €Ifl. One then has Proposition 1.5 :

functor pc ~ TG Proposition 2.5. The inclusion group G, ~ pc. Por any topo1ogiesl has a left adjoint R : TG R(G) - ltm GIN

subgroups of G with GIN (Ifl. where N ranges over al1 open normal

( -23-

Proof: We ooly need to verify that the canonica1 1\ map n: G ~ G has the universa1 property of a front

adjunction. The proof is ana10gous to that of Proposition 1.5

and Proposition 2.1.

Coro11ary. PC is a full ref1ective subcategory

of TG •

Given any group G, we cou1d a1ways regard it as a

discrete topo10gica1 group. The fo11owing resu1t is an immediate

consequence of Proposition 2.5.

Proposition 2.6. PC is a reflective subcategory

of G. The ref1ector R: G ~ PC is described as fo11ows:

For any group G,

R(G) - l1m GIN

where N ranges over a11 normal subgroups of G with GIN Elfl.

Remark 2.7. Although PC is a ref1ective subcategory

of ~, in general, it is not full. To see this, take CaF and

let G -(Y2'f) )(~e the product of )(0 copies of ,!:!2b then tbe

restricted product K _('!:!2Y.'){0) is a dense subgroup of G.

( -24-

By considering G as a vector space over zl2~ one can find a basis B of G containing a basis B1 of K. Let H be the vector space spanned by B\{b}, where b€ B\B • Then H, being a proper 1 dense additive subgroup of the profinite group G, is not c10sed in G. The quotient map ~: G ~ G/H is then not continuous with respect to the discrete topo10gy on the two e1ement additive gtoup G/H, whence ~ is a morphism in ~ but not in PF. This counter examp1e is suggested by Professor

M.Barr.

Remark 2.8. Let G be a topo10gica1 group and e its pro-f-completion. Denote by t the fami1y of a1l open normal subgroups of G with G/N € Ifl, then the canonica1 map n : G ----+ a is injective if and only if N2~ N - the identity. We shall calI G residual1y f if n in injective. In particular, if f is the category of finite groups, finite p-groups, or finite solvable groups, then every free discrete group is residually f.

(see e.g.Hagnus-Karrass-Solitar [1]).

Let! be one of the following categories: PT op , Top,

FU, PF, TG, ,9., §. and let U: PC ----+ ! be the canonical functor as described in Propositions 2.1, 2.2, 2.3, 1.5, 2.5, 2.6 where -25-

it has been shawn that U has a left adjoint F : ! - PC. For

any X € I!I, we shal! cal! F(X) the free pro-f-group on X. Sometimes we write FC(X) instead of F(X) to specify the category f.

Definition 2.9. Let X be a set and M the filter of

complements of finite subsets of X. Then FC(X,M) will be called

the Serre free pro-f-group on the set X, and is denoted by sFC(X).

Proposition 2.10. For any set X and M the filter of

complements of finite subsets of X, the following statements are

equivalent.

(a) G is a Serre free pro-f-group on X. (b) G = l!m LIV where V ranges over aIl normal subgroups of the free discrete

group L on X, such that L/v € Ifl and V contains aIl but at most finitely many elements of X. (c) G is the free pro-f-group on the pointed

topological space (X li {*}, *) where X li' {*} is the one-point compactification of X viewed as a discrete space.

Proof. Let t be the family of aIl normal subgroups

V of L with LIV € Ifl, which contains al! but at most a finite

number of elemeots of X. Let M be the filter of the complements

of fioite subsets of X. If N is a normal subgroup of L vith

" - -26-

LIN € Ifl, then the following are equivalent:

(al) N n X € M

(b 1) N € t

(Cl) (aN) n (X Û {*}) is open for aIl a € L, where X Û {*} is the one point compactification of X. The free pro-f-groups of (a), (b), (c) are inverse limit of quotients LIN e Ifl, where N satisfies (al), (b ' ), (Cl) ab ove , respectively.

Remark 2.11. In partièular, if f - I, then for any set X, sFC(X) is precisely the free profinite group as defined in Serre [1].

Remark 2.12. sFf is not a functor from ! to pc.

Proposition 2.13. For any finite set X,

SFf(X) - FC(X)

~. X is finite, so every normal subgroup of

L(X) contains aIl but at most a finite number of elements of X and hence

sFC(X) - l!m L(X) IN - FC(X) where N runs through a11 normal subgroups of L(X) with

L(X) IN € Ifl.

------_._------27-

Theorem 2.14. For any non-empty discrete space X, let

F(X) be the free profinite group on X. Then )/ 2 0 if X is finite IF(X)I = { Ixi 22 if X is infinite Proof. Case 1. X is finite, say Ixi = n. Denote

by ~ the family of aIl open normal subgroups of the free di8crete

group L on X such that LIN € I.~I. Then F(X) = l1m LIN N€4I

IF(X)I 1: Il!m L/NI ~ ln L/NI s n ~ Nd 41 0

We want to show that 1411 ~ ~o' To see this, let ~j 1: set of

aIl normal subgroups of L of index j. If H € ~j' then H i8 free

on m - j(n-l) + 1 generators (see Magnus-Karrass-Solitar [1]. m Theorem 2.10, p.104). One can define an injection 4Ij ---+ L(X)

sending H to an m-tuple consisting of generators of H, and so

1 Lm 1 s )/ m 1: )./ o 0 Bence

I~I - I ~j s )./ .)/ • 'A' je! 0 0 0 which impl1es that )/ )/ IF(X) 1 s ')/ 0 2 0 0 - ln particular, we know that

IF(1) 1 - 1 1!Jn {fIn! : n e !}I )1 )/ • In z 1 ~ n )! • >1 0 • 2 0 p-p poo Consequently, we have I,(x) 1 - 2~o • -28- c

Case 2. X is Infinite. First, we assert that

ILl = Ixi. Let w(n) denote the number of words in L of length n (not necessarily reduced), then n w(n) = I(X u x-~nl a Ixnl - Ixl = Ixl therefore

w(n) = ~o .Ixl a Ixl

The reverse inequality i8 trivial and so 1LI.. 1xl. Let ~ be

the family of all normal subgroups N of L with (L : N) < œ

then

and so

IF(X) 1

It remains to establish the reverse inequality. Hewitt [1] has

shawn that for any 1nfinite cardinal number c, the compact Hausdorff c space 1 2 (where 1 denotes the dis crete space {O,l})contains

a dense subset of cardinality c. Applying this result to the

Infinite discrete space X, it follows that there is a continuous 21xl 21xl map g : X - 2 whose image i8 dense in 1 . By viewing 2 as a dis crete topological group, it turns out that 21xI 1 is a pro-2-group. Then there exists a unique continuous 2 1xl homomorphism h : F(X) ----+ 1 with h.nx - g where

l'\x : X ----+ F(X) 1s the canonical map. hF(X) is closed and -29- -

__ __ 2 1xI contains g(X). Then hF(X) - hF(X) ~ g(X) = 2 ,which implies - 2 1xI - that h is surjective and so IF(X) 1 ~ 2 Consequently 2 1x1 IF(X) 1 = 2 as desired.

For any topological space X, one has a canonical

continuous map nX : X----+ FC(X), Beside this, we have no other

information (e.g. injectivity) concerning the map nX' Certainly,

one could ask the following question: What is the necessary and

sufficient condition for nX to be injective? or even more -

an embedding? (By saying that nX is an embedding we mean the map

nX : X ----+ nX(X) is a homeomorphism). In particular, if X /1 is a discrete space, then FC(X) - L(X) , and so by Remark 2.8

nX is injective for ~ - category of finite groups, finite p-groups

or finite solvable groups. Later, we shall show that if X is

a compact space, then nX is an embedding if and only if X is a

Hausdorff, totally disconnected topological space.

Notations. Let X be a topological space and let

! - ~i 1 i € l} be a family of equivalence relations on X. Preorder

1 by i S j iff Rj c: Ri' For each i € l, let 1f i : X - X/Ri

be the quotient map. For any i S j in l, there is a continuous

surjective map -30-

defined by Wji(Rj(X» = Ri(X) where Ri(X) denotes a typical element in the quotient space X/Ri. Clearly we have w w - w for ji kj ki any i S j ~ k in I. 50 we obtain an inverse system {X/Ri' i € l, n }. ji For any space Z, each continuous map

(i ~ j in I) induces a map

given by nji*(g) ~ g 0 n • This in turn yields a direct system ji

{Top(X/R , Z), i € l, n *}. Let n * : Top(X/R , Z) ----+Top(X,Z) i ji i i be the map induced by ni' i € I.

Lemma 2.15. Let X be a compact Hausdorff space and

let R - {Ri : i € I} be a non-empty family of equivalence relations

on X satisfying the following conditions:

(i) X/R is finite and dis cre te for a11 R € R.

(ii) If 5 is any equivalence relation on X with

the property that X/5 1s finite and discrete,

then there èxists an R € R such that ReS.

(Ui) n {R : R € ~.l - {(x,x) € XxX}.

Then X is homeomorph1c to lj;m X/R. Moreover, for R€R any finite discrete space Z, one has a natural isomorphism -31-

Top (X,Z) ~ 1!Jn° {Top(X/R, Z) : RE!}

Proof. First we claim that the pre-ordered set l is directed. Let Ri' Rj E R, then there is an injection

X/RinRj ----+ X/Ri x X/Rj which implies that X/RinRj is finite. For every x E X, (RinRj)(x) = Ri(x) n Rj(X); since X/Ri and X/Rj are discrete, Ri(X) and Rj(X) are both open in X and so is their

intersection. Therefore (RinRj)(x) is open in X and hence

X/RinRj is discrete. By condition (ii) in the hypothesis, there

exists ~ E! such that ~~ RinRj , which then implies that

i S k, j S k, i.e. 0 1 i9 directed. From condition (i), it follows

that X/R is compact and Hausdorff for each RE!, and now

Proposition 0.9{c) implies that ljm {X/R : RE!} i8 non-empty.

Consider the continuous map g : X ----+ lim X/R induced by the +

quotient maps X ----+ X/R (R E !). By Proposition 0.10, g is

surjective. We shall show that g is injective. Suppose gx - gy

for x, y in X, then (x,y) E R for aIl RE! and so by condition (iii)

in the hypothesis, we have x-y. These show that g is bijective.

By Proposition 0.11, g is a homeomorphism. This proves the !irst

part of the lemma.

To prove the second part of the lemma, we proceed as follows.

For each Ri ( !' let gi : Top(X/Ri , Z) ----+ y be a mapping of sets, -32-

and suppose that for i S j in l, the diagram

y

TOp(X/R , Z) i c01IIIIutes. We shall show that there exists a unique map g : Top(X,Z) - Y such that g olT i * .. gi for a11 i € 1. Let h € Top(X,Z) and define an equivalence relation 5 on X by xSx' iff h(x) - h(x'), x, x' in X. Since h induces an injection hs : X/S - z, X/S ls fioite and discrete and so by condition (ii) in the hypothesis there exists Ri ~ ! with Ri C S. Define g : Top(X,Z) - Y by

g(h) - gi(hSo lT R S) i' where lT R 5 : X/Ri - X/S is ioduced by the inclusion Ri C S. i' We shall show that the definition of g is iodependent 00 the choice of Ri €!. Suppose Rj € ! and Rj C S. ,Then we have seen

€ Renee before that there exists ~ ! with ~ C RinRj • -33-

and this shows that g ls well-defined. We shall show that g is the map with the property 8 ° 1T 1* - gi for a11 i E: 1. To see this, let u E: Top(X/R , Z) and let h - 1T *(U). As before, let S be i i the equlvalence relation induced by h : xSx' iff h(x) = h(x') for all x, x' ln X. Then R c S, indeed if xRiy, then 1T (X) .. ni(y) 1 i and so h(x) .. (n *(u»(x) U(1T (X» - u(ni(y» S (1T *(U»(y) h(y). 1 = i i = Therefore

g(1T i *(U» = g(h) - gi(hSonR S) i' But hSonR S(Ri(x» = hS(S(x» ~ h(x) = 1T i *(U)(x) - u(ni(x» i' .. u(Ri(x» i.e. hS ° 1TR S = u, whence 8(1T1*(U» - gi(u). Hence gn i * • 8i • i' The uniqueness property of g follows immedlately from the fact Thus that ni * is surjective. Top(X,Z) " Hm' «op(X/R, Z) : R E: !} -+ and this isomorphism is natural in the following sense. Let X' be another compact Hausdorff space, and suppose that the hypotheses of this lemma are also satisfled for (X',!') in the place of (X,!>. Then every contlnuous map

a X-X' induces an injection a'1 X/S - X' /R'1 (R'i- € R') where S - {(xl ,x2) € xl (a(xl ) ,a(x2» € Ri} •

There exists an equlvalence relation Rj E: ! vith Rj C S and -34-

for every such Rj' the fo11owing diagram commutes:

Cl* Top(X' ,Z) • Top(X,Z)

o Cl' 0 n • This completes the proof of - u i j the lemma.

Proposition 2.16. Let X be a compact, Hausdorff,

totally disconnected topo1ogical space. Then there exists a

non-empty fami1y ~ of equiva1ence relations on X ordered by

inclusion and such that (i) X = l!m' {X/R : R € ~} where each X/R is finite and dis crete

(ii) For any finite discrete space Z, one

has a natural isomorphism

Top(X,Z) = l!m' ~(X/R, Z) : R € ~}

~.. Let ~ be the family of all equivalence relations on X such that X/R is finite and discrete. In view of Lemma 2.15,

it suffices to show that R is non-empty and satisfies the hypotheses -35-

of Lemma 2.15. Since every compact Hausdorff totally disconnected topological space is O-dimensional (see for example Hurewicz

& Wallman [1]), it follows that X has a base of both open and closed sets. Thus for any x ~ x' in X, since X is Tl' there exists an open neighbourhood N of x not containing x'. Since

X is O-dimensional, there exists a subset A of X, which is both open and closed in X and such that x € A c N. Define an equivalence relation R on X by:

aRa' iff either a, a' € A or both a, a' , A (a, a' € X).

Then clearly X/R has only two elements and is also discrete.

Therefore R € R. This proves that ! ~ 41. The above argument also shows that n{R : R € !} =' {(x ,x) E XxX}. Hence R satisfies the hypotheses of Lemma 2.15.

Lemma 2.17. Let G be a pro-f-group, and D a discrete set. A map

f: G-D is continuous if and only if there exists some open normal subgroup

V of G and a continuous map

f' : G/V ----+ D such that f • f' 0 n where n : G ----+ G/V is the quotient map.

Proof. Necessity. Let t be the family of aIl open normal 8ubgroups of G. G i8 compact and D i8 di8crete imply that

._------...... ~ ... _... -._ .. , -36-

fG is compact and dis crete and hence finite. Say fG =' {al, ••• ,a } n -1 and let Si = f ai. Then Si is both open and closed, for each i • 1, ••• , n. On the other hand, by Proposition O.B(d),

Si '" Si = N2~ SiN ,(i = 1, ••• , n). Every open subgroup is closed, therefore N is closed in G and ~being closed in G is compact. lt now follows from Proposition O.B(e) that each SiN is closed in G, i - 1, ••• , n, N ~ ~. For each i ~ U, ••• , n}, . C we then have a family {Si' SiN : N ~ ~} of closed subsets of C C G, where Si = G\Si. Now Si n ( n SiN) - ~ and N there exists il' ••• , i such that (*) S~ n ni

Let V=

Then V ri- ~ and V is an open normal subgroup of G. There is

an injection

ni and since f is closed under finite products, ini jUI GlN i ~ Ifl j

and so does its subobject G/v. Therefore V ~~. Define

f' : G/V ----+ D

by f'(xV) • fx. We shall show that this is well-defined. To

see this, first we claim Si • SiV for aIl i • 1, ••• , n. -37-

Indeed,

C Si (by (*»

Suppose xV - yV in G/V, then f'(xV) - ai for seme i, and so - -1 x E: f {ai} - Si,Since SiV - Si it follows that x E: SiV and hence

xV C Si' Therefore yV = xV C Si' In particu1ar y E: Si and f' (yV) = fy = ai' whi,ch proves that f' (xV) - f' (yV) as desired, C1ear1y f - f'n, From the definition of the quotient space,

f' is continuous if and on1y if f - f'n is continuous, Thus f'

is a we11-defined continuous map, The sufficiency is obvious,

Let' {G , i € l, f ji} be a directed inverse system of i pro-f-groups and let G 1jm G • For any dis crete group D, = i id the fji's induce a compatible fami1y of maps,

We then have a direct system

Let

, be the map induced by the canonica1 projection Pi : G - Gi It fo11ows that there exists -38-

a unique map

f : 1im PC(Gi ,D) - PC(G,D) making the fo11owing diagram

f PC(G,D) l!m PC(Gi ,D ) ----- ~ commute, where t is the obvious map determined by the direct i limit.

Proposition 2.18. Asswne the same notation as above.

If each f is surjective, then f is bijective. ji Proof. f is injective.

From Proposition O.9(c), for each i € l,

j € I}

8y hypothesis, f ji is surjective for a11 i S j in l, and so

Pi(G) -Gi which proves that Pi is surjective for each i € I.

We c1aim that each Pi* is injective. Indeed, if Pi *gl -Pi *g2 then glPi - g2Pi whence gl - g2 since Pi is surjective. So

each P 18 injective and hence f 18 injective. i * -39-

f is surjective.

Let g € PC(G,D). Then by Lemma 2.17, there exists an open normal subgroup V of G, G/V €lf1and a continuous homomorphism g' : G/V ----+ D such that

1T Il' g = (G ----+ G/V ~ D)

-1 By Proposition O.7(b), V ~ Pi Ni for some open normal subgroup

Ni of Gi with Gi/Ni €Ifland i € 1. The canonica1 projection

Pi : G ----+ Gi then induces an injection pi : G/V ----+ Gi/Ni ' which is al80 surjective since Pi is.

De fine ID hi : Gi/Ni , ,-1 by hi(xNi ) = g (Pi (xNi »

C1ear1y, this is continuous and hpi - g' •

De fine mi : G ----+ D as the composition 1 1T h Gi~G/Ni i ~D where 1f 18 the quotient map. Let i

We shal1 show that fm • g which then imp1ies that f ls surjective.

- miPi - hi lT 1Pi

hilTiPi(x) - hilTi(Pi(x» - h1(Pi (x)N i )

- g'(pi-1(Pi(X)Ni » - g'(xV) - g'(nx) • (g'lT)(x) - g(x) -40-

Proposition 2.19. Let X be a compact Hausdorff totally disconnected topological space, and let! be a non-empty family of equivalence relations on X satisfying the hypotheses of Lemma 2.16. Then F (X) = l!m F (X/R) where the inverse limit is taken in PC, with respect to the maps F(X/R ) ----+ F(X/R ) induced by X/Rj ----+ X/R for j i i

Rj C Ri in R. Proof. Let G be a pro-f-group then G lim GIN = -4- where N ranges over a11 open normal subgroups of G. We know that PC(l!m F(x/R), G) = l!m PC(ltm F(x/R), GIN) Binee the functor (ltm F(X/R), _) : PC - !. preserves inverse limits. One, therefore, has PC(lj;m F(X/R), G) = lj}n PC(l!m F(X/R), GIN) R N R

:: lj;m 1,1m PC(F(X/R), GIN) (by Proposition 2.18) N R :: lj}n l!m Top(X/R, GIN) N R :: lj;m Top(lj;m X/R, GIN) (by Proposition 2.16) N R -41-

== lim Top(X, GIN) N

== 1im PC(F(X), GIN) + - N

== PC(F(X), l!m GIN) N

== PC(F(X), G)

Thus l!m F(X/R) == F(X) R

Coro1lary. Let G be a pro-f-group and Fc(G) the free pro-f-group generated by the underlying topo1ogica1 space of G. Then

where N ranges over the family ~ of aIl open normal subgroups of

G.

Proof. Every N € ~ induces a congruence relation

~ on G: -1 ~x' iff x x· E: N for x, x' E: G.

Let! - {~ 1 N E: ~}. We shall show that! satisfies the hypotheses of Leuma 2.15. Let S be an equivalence relation on G such that

GIS is finite and discrete. For each a E: G, there exists an open normal subgroup N E: t 9uch that aN c lb : bSa}. Now, a a

{aNa : a E: G} i9 an open cover of G, so there exists a finite -42-

(

n subcover {aiN i = 1, •.• , n}. Let N = i~h Na and R = ~ ai i be the congruence relation on G induced by N. If (x,y) € R, -1 then x y € N and so xN - yN. On the other hand, x € aiN , ai for sOlDe i, jE:' U, ••. , n} and so (x,a ) E: Sand y E: ajN a i j .(y,a ) E: S, xN c aiNaiN .. aiNai c S(a ). Simi1ar1y yN c S(a ), j i j which imp1ies that S(ai) n S(a ) $. Thus S(a ) = S(a ) and j + i j

(x,y) E: S. So we have ReS, which proves condition (ii) of

Lemma 2.15. Condition (iii) i8 trivial since N2~ N is

the identity. Since GIN ~ G/~, condition (i) is c1ear1y

satisfied. Thus by Proposition 2.19,

Theorem 2.20. Assume that C - 1 . Then fo"r any compact space .. X, the canonica1 map

is an embedding if and on1y if X is Hausdorff and tota11y

disconnected.

Proof. Necessity

'CCX) i8 Hausdorff and tota11y disconnected and since

these properties are hereditary, X being homeomorphic to a

8ubspace of 'CCX) also has these properties.

_____oo._ •• -.--.----,:-~.-C".'7 CC" • -43-

Sufficiency

Assume that X is compact, Hausdorff and tota11y disconnected. Let! be the fami1y of aIl equiva1ence relations

R on X for which X/R is finite and discrete. By Proposition 2.19, l(X) = 1im l(X/R). We then have the fo11owing commutative R diagram,

X • j X/R l(X/R)

where w, w' are canonica1 maps. Since X/R is discrete, it fo11ows that / is injective for aIl R € and this imp1ies nX R !' that nX i8 injective. Indeed if nX(x) - nx(x'), then n'nx(x) = wnx(x'), The injectivity of and so nX/R n(x) - nX/Rw(x') for aIl R € !. n / imp1ies that R(x) • R(x') for aIl R € R. Since X = 1im X/R, X R ~ we must have x - x'. -44-

A topological space is said to be strongly zero-dimensional if for each closed subset A of X and each open neighbourhood U of A, there is an open and closed neighbourhood of A contained in U.

Corollary. If X is a strongly zero-dimensional

Hausdorff topological space, then the canonical map

is injective.

Proof. Consider the adjoint pairs,

FC---t U PC - Top

B ---f U CHTop - CRTop where B is the Stone-~ech compactification functor. For any X € 1CRIop l, and G € 1PC l, one has

(FC(X), G) ~ (X, UG)

~ (B(X), UG)

which ~plies that

FC(X) ~ FCB(X)

--_._------.. _-----_.. _---_.... -45-

Since X is strongly zero-dimensional, it follows that BeX) is totally disconnected (see e.g. Bourbaki [1], part 2, p.258).

By Theorem 2.20, the map

is then injective. But nX equals the composite map h nS(X) X -----+ B(X) -----+ FCB(X) where h is the canonical embedding. Therefore nX is injective. -46-

r,

CHAPTER 3

SUBGROUPS OF FREE PRO-C-GROUPS

Serre [1) has shown that c10sed subgroups o! Serre

free pro-p-groups are aga in Serre free pro-p-groups. In this

chapter, we investigate certain c10sed subgroups of free

pro-f-groups. We sha11 show that the sma11est c10sed normal , subgroup of FC(XU{*}) .( the free pro-f-group generated by

the coproduct of a compact Hausdorff to.ta11y disconnected tOPQ1ogica1 space X and a single point) containing * is again a free pro-f-group. Furthermore, we sha11 prove that the

Sylow p-groups G of free profinite groups generated by a p topo1ogica1 space X are Serre free pro-p-groups and we

obtain a formula for the minimal number of generators of

G for the case when X is d~rete. p

Theorem 3.1. Assume that f ;t l is c10sed under group extensions. Let X be a compact, Hausdorff, totally

disconnected topologiea1 space; FC(X), the free pro-f-group

generated by X ; and FC(FC(X» the free pro~-group generated

by the underlying topologiesl spaee of FC(X). Let X~~} be the coproduct in Top of the topologica1 sp8ce X and a single

point *. One then bas a split exact sequence in PC

_.. _------:- -47-

1

where S is induced by

Xu {*} ( x -+ [x], * -+ 1)

y ia induced by the canonical injection

x ---~I Xü{*}

and a is induced by -1 Fe (X) ( a -+ y(a) [*]y(a».

(The square bracket [ ] is used to denote the canonical embedding

nX : X -+ Fe(X) (x -+ [x]) as given in Theorem 2.20.)

Proof: For each open normal subgroup N of Fe(X) ,

we let ~ be the semidirect product of Fe (Fe (X)/N) by Fe(X)/N with respect to the action

defined by

- F e (8 N'(t» where eN'(t) is the bijection -1 FC(X)/N ----+ Fe(X)/N (s -+ st )

If N' i8 aoother open nor.al subgroup of Fe(X) and NcN', then

one has a commutative diagram, with split exact rows:

, i -48-

aN SN 1 • FC(FC(X)/N) .~ FC(X)/N • 1 YN

SN' F (X)/N'l ----4 1 1 l l • FC(FC(X)/N') a--+~' 1 C -- N' Y , N

We app1y 11m, and using the Cor011ary of Proposition 2.19, we obtain a split exact sequence:

1 _a_o+. H y where H = 11m~. Since f is c10sed under group extensions, each ~ is a pro-C-group and so is H = 1~m HN • One has the f0110wing commutative diagrams:

a S 1--.. IH t F (X)- 1 • Ff(Ff(X» C y lF('N" l'N l'N' aN SN 1--.. 1 FC(X)/N ___ 1 FC(FC(X)/N) --- ~ 1 YN

We shall show tha t

in PC and the maps a t St Y satisfy the statementa of

this theorem. -49-

Define li: Xù !*} ---) H as follows:

1 y([x]) if X€X li (lt) = L a([l ex)]) if x = * Fc

Then li is con tinuous. To show tha t H == Fe (XU {*}) i t

suffices to verify that li is universa1 in the sense that to

each continuous map ~ : Xu {*} -+ K ,K€ IPCI, there exists

a unique continuous homomorphism $ : H -+ K with ~ll = ~.

The map ~Ix: X -+ K induces ô: FC(X) -+ K and we can

define a : FC(FC(X» -+ K to be the map induced by

---_1 K (a -+ ô(a)-lH *)ô(a) )

Note that every e1ement of H can be written unique1y in the form

h .. a(c)y(d)

Indeed,

-1 -1 whence c-a (h·yB(h », d-B(h), and if a(c)y(d)-a(c')y(d'), then

d - Ba(c)By(d) - Ba(c')By(d') - d' ,

so that a(c)-a(c') and hence cac' since a 1s injective. -50-

Define

K as follows:

~(h) = a(c)oô(d) where h=a(c)oy(d). Then ~ is continuous sinee it is the composition of the following continuous maps:

H

h 1+ (c=a-1 (h·yB(h-1 »,d-B(h»1+ ( a(c),ô(d) )1+ a(c)oô(d) where is eontinuous by Proposition 0.11.

We proceed to verify that ~ is a homomorphisme Let

h • a(c)y(d) , hl = a(cl)y(dl )

be elements in H, c,cl ~ FC(FC(X», d,dl ~ FC(X). Then

hh - a(c)y(d)a(c )y(d ) 1 1 l • a(c) (y(d)a(c )y(d-1 »y(d)y(d ) 1 l -1 Observe that y(d)a(cl)y(d ) ~Ker S· lm a and since a is

injective there exists a unique e~Fc(Fc(X» such that

-1 aCe) - y(d)a(c )y(d ) • 1

Therefore, hh - a(c)a(e)y(d)y(d ) 1 l

-'."'"..,...... ,.--.-_._---_. -51-

and so

On the other hand,

Thus to show that ~(hhl) = ~(h)~(hl)' we need only to verify that

cr(e)ô(d) ô(d)cr(c ) • = l

Since Fe(Fe(X» is generated by Fe (X) , we may assume, without loss in generality, that cl-[cOl for some Co in Fe(X); due to the fact that both cr and the map

K

would lead to a commutative diagram:

Fe (X) K -52-

So one has

-1 = YNnN'(d)oaNF(nN')([cO])oYNnN'(d ) -1 = YNnN'(d)oaN([nN'cO])oYNnN'(d )

l ]) • aNFf(nN') ([COd-

-1 • nNa([cOd ]).

, This being the case for every open normal subgroup N of FC(X) we conclude that

so that one bas -1 a(e)6(d) • a([cOd ])6(d) • Ô(COd-l)-lQ(*) Ô(COd-l)ô(d) -1 9(*)Ô(C ) • 6(d)ô(cO) O • ô(d)a([cO»'

-----_.----.. _---. - ....• _- .. -53-

This proves that ~ is a homomorphism. For each xeX, one has

H\J(x» = 1jJ{y([x]» = ô([x]) = $(x)

H\J(*» .. 1jJ{a([l (X)]» = a([~ (X»)) . F c c -1 = ô(~ (X» $(*)ô(l (X» = $(*). c Fc

which proves that ~\J. $ and it remains to show that ~ is unique with this property. This is immediate if we cou1d show that lm \J generates H. To see this, let aeFC(X)/N, then

(**)

-1 a (l,a ). ([~ (X)/N],l).(l,a) C

• ([a],l)

which imp1ies that Hw is generated by ~([~ (X)/N1) and C

YN(n~FC(X». i.e. ~ is generated by nN(a[~ (X)1) and C wN(yFC(X». If we let G stand for the closed subgroup of H generated by a(lFc(X)1and y(Ff(X»; (i.e. by \J(*) and \J(X» then wN(G) • ~ for aIl open normal subgrou~N of Ff(X). -54-

lt then fo11ows from Proposition 0.10 that the maps

G ~ ~ induced by the maps n : H ~ ~ yie1d a N surjective map G ~ H which is impossible un1ess G=H.

Thus H is generated by~(X)u~(*) = lm ~ • This completes the proof that H =Fc(XU{*}). From the universa1 property of Fe (X~ {*}) we know tha t the isomorphism Fe (xu {*}) = H is induced by the map ~ : Xu~} ~ H making the fo11owing diagram commute :

Xu {*}

H

Since ~(x) • y([x]) for a11 x€X, and ~(*) a a[l (X)] ,we cou1d F C make the fo1lowing identifications:

= H

[x] _-__1 y([x]) (xeX)

[*]

With these identifications, we have the following split exact sequence, -55-

a 1 Fe(XÜ {*}) ~ Fe(X) - 1

Y

Let a€Fe(X), then from (**) we have,

This 1s true for aIl open normal subgroups N of Fe(X) , so one has

-1 a([a» - y(a )'a([~ (X»)·y(a) c

-1 • y(a )·[*]·y(a).

Bes1des, for any x€X,

S([x]) • S(y[x]) • [x]

S([*]) • Sa([lFc(X)]) • IFe (X) y([x» - [x] The proof of the theorem 1s now complete •

.. _u_ ..... __'.-.._. __ .'· ______..•. ~ :-.- ".-. -56-

Remark 3.2. From the definition of the map , a : Ff(Ff(X»-+ Fe(XU {*}) , it fo11ows that aFe(Fe(X» is contained in N, the smallest closed normal subgroup of

Fe(XÜ {*}) containing [*]. On the other hand, the map B sends [*] to the identity element of Fe(X) and so B maps N onto the identity element of Fe (X) , i.e. NcKer B. By the exactness of the sequence aFe(Fe(X» = KerB , therefore aFe(Fe(X» coincides with N. Since a is itüective and continuous, by Proposition 0.11,

Fe(Fe(X» is homeomorphic to N, i.e. Fe(Fe(X» is homeomorphic to the smallest closed normal subgroup of Fe(XU{*}) generated by *.

eorollary. Let F(x,y) be the free profinite group

generated by two letters x,y. Then the smallest closed normal subgroup of F(x,y) generated by y is homeomorphic to the free profinite group F(!) generated by the topological space, underlying î. Proof: From Theorem 3.1, one has a split exact sequence,

1 F(F(x» _...;;a;;"-'_I F(x,y) -!- F(x) - 1 .....l- Heré F(x):! and so the result follows from Remark 3.2. -57-

Following Serre [1], we have the following definitions:

By a supernaturalnumber , we mean a formaI product

n = JI pn(p) p where p ranges over the set of aIl primes and where each n(p) is a non-negative integer or +œ •

Let G be a pro-f-group, and H a c10sed subgroup of G.

Then the index (G:H) of H in G is defined by

(G:H) = l.c.m. (GIN: H/HnN) where N ranges over aIl open normal subgroups of G.

The order ord G of G is defined by

ord G oz (G: 1)

Let p be a prime, a closed subgroup H of a pro-~group

G is called a Sylow p-group of G if H is a pro-p-group and

(G:H) is prime to p.

Theorem 3.3. Let X be a topologica1 space, F(X)

the free profinite group generated by the space X, and G a p Sylow p-group of F(X). Then

G = sF (')( ) p p for sOlDe cardinal >l where SF p (>1) denotes the Serre free pro-p-group

on')(generators. If X is dlscrete, thenÀ'satisfies the followlng:

." _o••• ,,~._ ••-.. __ • ____ -58-

(i) )! = 1if Ixi = 1. )( )( (ii) )/ S il S 2)(0 and 2 2 0 0 .. if 1 < Ixl

(Ui) }/ 2 1xl 2 = 2 if X is infinite.

Proof. Let

1 - A - B ~ F(X) - 1 be an exact sequence of profinite groups and continuous homom- orphisms, with A a pro-p-group. By Serre [1] (Proposition 1, p" 1-2) there exists a continuous section a : F(X) ~ B (i.e.

COO- idoF(X»,which composes with the front adjunction X~F(X) induces a homomorphism T:F(X)~ such that C.T = id.F(X)" It now fo11ows fram Serre [1] (Chapter I, Proposition 16, Coro11ary 1 of Lemma 4, and Coro11ary 2 of Proposition 24) that

G :: SF Of) p p for some cardinal )( • Case 1. Ixi - 1. We knew that F(l) :: lim --Z!Dl • -~ ...n Therefore G :: p

Case 2. 1 < Ixi < CD. For every integer n, consider

the wreath product

wr (y~) wn - (yp!:.> -59-

~- .. -

One then has a split exact sequence

1 -+- (Z/pZ)n -+- W -+ Z/nZ -+ 1 -- n-- - Since W is generated by two e1ements, there exists surjective n continuous homomorphisms

F(X) -+- W (n € li) n But for every positive integer n not divisible by p, the Sylow

p-group of W is of the form (!/p!)n. By Serre [1] (Proposition 4(b), n p. I-5), one has surjective maps

(*) G -+ (Z/pZ)n (n € (n,p) 1) p !' = Let H* stand for the intersection of a11 open normal subgroups

of H with index p. Then «!/p!)n)* is trivial, so by Serre [1]

(Proposition 23 bis, p. I-35) there is a we11-defined surjective

map, for each n € !, (n,p) • 1,

Gp /G* p -+ ('!:./ p!) n

therefore IG /G*I ~ p~o p P By Theorem 2.14, >t ISF p (i)1 • IG p 1 s IF(X)I • 2 0 ~ and so >1 s 2 0 •

By Serre [1] (p. I-35), one has

..... _.... _------60-

This implies that

>1 'A' p o 1 1 >1 s 2 0 s Gp /G'" p = P s IG p 1

){ i.e. 2>1 = 2 0 >t )/ Henee we have)( s 2 0 and 2)( = 2 0 • Clearly V o )/ sinee 2>1 = 2 0 •

Case 3. lx 1 ~ )/ • Consider X as an infini te 0 diserete spaee, it then follows from Hewitt [1] that there is a eontinuous map 21xI \ : X-2 1xI whose image is dense in 12 .

Besides, there is an obvious embedding -2 =' {0,1l - --Z/pZ .. {O, l, • •• , p-l} whieh induees a eontinuous map Ixl Ixl a : 22 . - (z/pZ)2 - -- 2 1xl whose image generates the abel1an pro-p-group (!/p!) •

The map Ixl a 0 \ : X - (Z/pZ)2 ls dense and so lnduees a surjective map Ixl y F(X) _ (Z/pZ)2 -61-

because lm y is closed and hence con tains the generators

21xI 21xI e(1 ) of (~p~) • Thus by Serre [1] (Proposition 4(b), 21xI p. 1-5), G is mapped onto a Sylow p-group of (~p~) , which p 21xl is clearly (~/p~) itself • Thus we have a surjective map

Ixl G /G* ----+ (Z/PZ)2 p P - -

Consequently by Theorem 2.14, and Serre [1] (p. 1-35) one has

1xl Ixl 2 ')t .. 22 2 s IG p /G*I p - p s IG p 1 s IF(X)I ){ 21xl i.e. 2 .. 2

The proof of the theorem is now complete.

The following result solves partially an exercise in Serre [1] (p. 1-38). We could not prove that ~ • 21xl unless we assume the generalized continuum hypothesis.

Proposition 3.4. For any infinite set X,

Fp(X) :: SFp(À' ) >t for some cardinal >t satisfying 2 where

F (.) and SF (.) denote the free pro-p-group and Serre free p p -62-

pro-p-group respective1y. Proof. In Serre [1] (Coro11ary 4 of Proposition 24, p. 1-37), it bas been shown that

for some cardinal )/. The rest of the proof is exactly the same as in case 3 of the proof in previous theorem, except for rep1acing Gand F(X) by F (X). P P -63-

CHAPTER 4

PRO-NILPOTENT GROUPS

Let G be a pro-C-group. Define inductively

G G, G !G , G] l = HI = i for every positive integer i, where , denotes the smallest IGi G] closed normal subgroup of G containing aIl the commutators (x,y),

X € G , Y € G. In fact [G , G] is the closure of (G , G) in G, i i i where (G , G) is the commutator subgroup. The family (G ) i i

is called the lower central series of the pro-~-group G.

Definition 4.1. A pro-C-group G is nilpotent if

G • 1 for some positive integer n. If n is the least positive n integer for which G - l, we say that G is nilpotent of class n. n

Proposition 4.2. (a) A closed subgroup of a nilpotent

pro-~-group (of class n) is nilpotent. (of class Sn).

(b) A continuous homomorphic image

of a nilpotent pro-~-group (of class n) is again nilpotent (of class

Sn).

(c) Direct product of nilpotent

pro-f-groups of class at Most n is nilpotent of class at Most n. -64-

(d) The nilpotent pro-f-groups of c1ass at most n form a variety in the category of pro-f-groups.

Proof. (a) Let H be a c10sed subgroup of the nilpotent pro-f-group G. Let (Hi) (respective1y (G be the i » lower central series of H (respective1y G). To show that H is nilpotent, it suffices to show that Hi C G for a11 i. We do this i by induction on i. For i • 1 it is c1ear1y true.

Hi+1 = [Hi' Hl C [Gi , G] -Gi+1 which proves (a).

(b) Let f : G ----+ G' be a surjective map in pc. Assume that G is nilpotent. To show that G' is nilpotent, we proceed to show that fG i '" Gi' where (Gi ) , (Gi) denote the lower central series of G, G' respective1y. Clearly we have

.. G' th Then fG l • fG • G' 1 • Assume that it is true up to i step. fG + • f[G , Gl. Since f is a c10sed, continuous map i l i f(G , G) - f(G , G) i i and so

fG i+l • f[Gi , Gl • f(Gi , G) - f(Gi , G)

• (fGi , fG) • [Gi ' G'l • Gi+l • (c) Let G - dh Ga be a product of nilpotent

pro~-groups. Denote (Ga,i), (Gi ) to be the lover central series

of Ga , G respective1y. We shall show that Gi • afi Ga, i for every positive integer i. This is done by induction. Clear1y -65- r it is true for i = 1.

[G , a G] Gi+1 = i G] [IIa Ga, i'

II G ] = [IIa Ga, i' a a

II G ) II = (IIa Ga, i' a a = a (G a, i' Ga )

= aII (G a, i' Ga )

The 1ast equa1ity fo11ows from the fact that in a product space,

the c10sure of a prodl1ct of sets i8 the same as the product of

their c1osures.

(d) This fo11ows fram (a), (b) and (c).

Proposition 4.3. Every nilpotent pro-f-group is

pro-ni 1po ten t •

Proof. If G is a nilpotent pro~-group then by

previous proposition, GIN is nilpotent for every open normal

subgroup N of G. Henee the resu1t fo11ows.

Proposition 4.4. Let f G ----+ Gr be a morphism in pc. Then

f(Ker f • [A,B) • [fA, fB)

for any subgroups A, B of G containing Ker f. -66-

Proof. C1ear1y f(Ker f • (A,B» .. (fA, fB). By IA,B]) Proposition O.8(e) Ker f • [A,B] is c10sed and 50 f(Ker f • is c10sed. Therefore

f(Ker f • [A,B]) ~ f(Ker f • (A,B»

= (fA, fB)

= [fA, fB]

On the other hand, since f is continuous, one has

f(Ker f • [A,B]) C f(Ker f • (A,B»

C f(Ker f • (A,B»

.. (fA, fB)

= [fA, fB]

This proves the resu1t.

Coro11ary 1. For any pro-f-group G,

(G/Gn)k .. Gk/Gn for a11 positive integers k S n. Proof. We prove this by induction on k. For k .. 1,

it is obvious. Assume that the resu1t is true for k-1 ~ O.

G/G as the canonica1 quotient map. Then Consider f : G - n fram Proposition 4.4, one has ) - f(Ker f • [G _ , G]) Gk/Gn • f(Gn.Gk k 1 - {(G/G )k_1' G/G ] • [fGk_1, fG] n n - (G/Gn)k which completes the induction process.

---'------'------'----,­. ,",- -67-

Corollary 2. For any pro-C-group G, GIG is nilpotent, n for every positive integer n.

Proof. This i9 a consequence of the above corollary, by taking k = n.

... Corollary 3. For any pro-f-group G, GI i~l Gi is pro-nilpotent.

Proof. By Corollary 2, GIGi is nilpotent for every positive integer i, and 50 by Proposition 4.3, GIGi is pro-nilpotent.

Therefore l!m GIGi is pro-nilpotent. The map

-; : G - l,!m GIGi induced by the canonical projections

G 1 GIGi is surjective, by Proposition 0.10, and the kernel is obviously ...... m i~l Gi • Bence GI i~l Gi and li GIGi are homeomorphic as pro-f-groups, the result then follows.

Corollary 4. For any pro-f-group G, G 1s pro-nilpotent ... if i!ll Gi - 1- Proof. This 18 an iDlDediate consequence of Corollary 3.

Leuaa 4.5. (a) Every pro-p-group is pro-nilpotent. (b) For every prime p, every pro-nilpotent

..._ .. _._~------~ -68-

group has a unique Sylow p-group.

Proof. (a) This is obvious since every finite p-group is nilpotent.

(b) Let G be a pro-nilpotent group, and

~ the family of aIl open normal subgroups of G. From the theory of finite nilpotent groups, for every N ~ ~, GIN has a unique

Sylow p-group say SN • If N c N' in ~, the canonical map GIN - GIN' ,p maps SN,p onto SN',p. Then Sp = l!m SN p is a Sylow p-group of N~~ , llm GIN. From Proposition 0.12, G = llm GIN, so by identifying G N~~ with 1!m GIN, it fo1lows that S is a Sylow p-group of G. In N~~ p fact S is unique, for if S ' is another Sylow p-group of G, then p p

the quotient map G ----+ GIN sends S ' onto a Sylow p-group of p GIN, which must be SN • Therefore S'=llmS -S. ,p P N,p P

Theorem 4.6. The fo1lowing statements are equivalent

for a pro-f-group G:

(a) G is a pro-nilpotent group

CD (b) i!ll Gt - 1 (c) G is the direct product of tts Sylow

p-groups, one for each prime p.

Proof. (6) implies ft)

Let ~ be the family of all open normal subgroups of G. For each -69-

N € ~, consider the canonical quotient map n : G ~ GIN. N It has been shawn in Proposition 4.2(b) that nN(G ) (G/N)i ' i = for every positive integer i. GIN is nilpotent implies that for some integer n (G/N)n = l, i.e. nNGn = 1. This means that N :> G , and so n co i!h Gi C N2~ N = 1 (b) implies (a)

This is the Corollary 4 of Proposition 4.4. (a) implies(c)

For each N € ~,G/N is finite and nilpotent, it then follows from the theory of finite nilpotent groups that GIN is a direct product of its Sylow p-groups, one for each prime p, say

GIN - TI N P P where N is the Sylow p-group of GIN. (The uniqueness of N p P follows from the fact tbat every Sylow p-group of a finite nilpotent group is normal). The canonical map

lTN,N' : GIN ~ GIN' (N eN' in ~) sends N onto N'and so one has a compatible family p p

{N : N € t}, for every prime p. We shall show that p G::TIlj;mN p N P

We proceed to show that TI lim N has the univers al property of p N p -70-

.--, 1

the inverse limit of {GIN, NE:~ } • The canonical projections

1I : lim N - N (p prime) N,p N p p

induce a continuous homomorphism

1I : il lim N il N = GIN N p N p P P making the following diagrams commute,

GIN

(N c N' in ~)

GIN'

Suppose there exist morphisms ~: H - GIN, N E: ~ in PC, such

that fN1rN,N' • fN" Then we have a compatible family of morphisms

IH-I--. N (N € t) defined by the composite maps P H • GIN .. il N N (N E: ~ ppp~ ~)

where 11 is the projection map. So for each prime p there exists p a unique morphism

a H --_. 1!Jn N P N P

such that 1r N,p.ap • 1rp.fN" The a 's then in turn induce a unique morphism p -71-

a H --...., II lim N P N p making the following diagrama commute,

a H JI lim N p N p

'IT p* (for each prime p)

where 'IT ia the canonical projection. p * One then has the following commutative diagrams,

a H JI lim N p N p

GIN

Indeed,

'IT 'IT a - 'IT 'IT a p N N,p P*

- 'IT N,p a P - 'IT P .f N

whlch 15 true for aIl primes p.

lt remains to show that Q la unique with thls property. Suppose -72-

there exists a morphism

a' H TI1im N ---_1 P +- P N for a11 N €~. Then with ~N'al = f N

~ ~* a' = ~ ~ a' = ~ f = ~ a = ~N n*a • N,p P P N P N N,p P ,p P

one has n*a , but a is This is true for a11 N € ~ and so p proves unique with this property, hence a = a' as desired. This that G = lim GIN = TI 11m N N p N P follows. and since 1i m N is a Sylow p-group of 1im GIN, the result i p N

(c) implies (a)

By (c), G is a product of pro-p-groups and so by Lemma 4.5

G is pro-nilpotent.

,T , •••• ,T }} denote the non-commutative ting Let ~{{ Tl 2 n , •••• ,T with of formai power series in n indeterminates Tl ,T2 n in Z • Elements of Z {{ Tl' T , .... ,T }} can be coefficients -p -p 2 n ---+ Z where M i6 the free discrete regarded as functions K -p ••• , T n } } mono id on Tl' T2' ••••• T n ; and so one can endow ~ {{ Tl' T2' with the topology of pointwise convergence 60 that

...... T }} = ~ as topological spaces. ln Lazard [1], ~ {{ Tl .T2 n -73-

it has been shown that the closed multiplicative subgroup of

~ {{ T ,T , •••• ,T }} generated by l+T , l 2 n l 1+T2,····,1+Tn is a free pro-p-group on n generators. Replacing Z by ~ , " - one has the following

Theorem 4.7. Let K be the closed multiplicative

subgroup of! {{ T ,T , •••• ,T }} generated by 1+T ,1+T , •• l 2 n l 2 ••• ,1+T , then K is a free pro-nilpotent group on n generators. n Proof. Let (K ) be the lower central series of K,and n M the free discrete monoid generated by T ,T , •••• ,T • Define l 2 n

by

ô(f) = f-l Clearly ô 1s continuous. Since ô(fg) = fg - 1

• f(g - 1) + (f - 1)

• ôf + fôg ô(f,g) _ ô(f-lg-lfg) _ f-lg-lfg - 1

_ f-lg-lôfôg _ f-lg-lôgôf -74-

it fo11ows by induction that one has n o(K _ ,K) c I n 1 where I is the kerne1 of the augmentation map

---)~ which sends every forma1 power series onto its constant term

(i.e. f ---+ f(Q). Since 0 is continuous, we have

c o(K _ ,K) n 1

M But In is c10sed in! since it equa1s n {p-1 ({ 0 }) : deg m < n} m where deg denotes the degree of e1ements in M as defined in Lazard [1] M (p. 53), and p : ---+ are the canonica1 projections. Therefore m -~ -~

œ n œ and since ~lI = 0 it follows that n~lKn - 1. The group K i8 compact, Hausdorff and tota11y disconnected since

is, 80 by Theorem 4.6, K i8 pro-nilpotent. Consider the continuous M homomorphism a : ~

induced by the canonica1 projection ---+1 Z ~ -p -75-

M M 5ince both and Z are compact and Hausdorff, it fo110ws -2 -p that a is c10sed and so a maps K onto F , the c10sed p multiplicative subgroup of ~ H T ,T , ••• ,T }} generated by 1 2 n 1+T ,1+T , ••• ,1+T • By Lazard [1], F is the free pro-p-group 1 2 n p on n generators. Let

a : K 1 F p P be the restriction of a on K. C1ear1y a is a continuous p homomorphisme Let F be the free pro-nilpotent group on n

generators say x ' •• • Then by Theorem 4.6, 1 ,x2 ,xn F = rr 5 p P where 5 is the (unique) 5y10w p-group of F, one for each prime p. p Let

6 : {1,2, ••• ,n} 1 F

be the canonica1 injection ( i.e. 6(i) = xi' i = 1,2, ••• ,n.) Then we have a map

Bp { 1,2, •••• ,n} ---15 P

defined by 6 • n 0 where n F ---+ 5 1s the canonica1 p p B p p projection. This imp1ies a unique morphism

Bp* : F ---PI 5 p p with the property e* n - 6 ,where n : {1,2, •• ,n} - F ppp P P i8 the canonical injection.

". _._._._._--_. -76-

The composite map s Cl K PI F P induce a continuous homomorphism

À K II S '" F P P Then one has

lf À(1+T ) = e* a (1+T ) p i p p i

= e* (I+T ) p i

= epnp(i)*

= e (i) p

This is true for aIl primes p, so we have

for aIl i - 1,2, •••• ,n ; and this implies that K is isomorphic

to the free pro-nilpotent group on n generators. -77-

CHAPTER 5

TRIPLEABLENESS

Let F ~ U : ! -+- ! be an adj oint pair. Eilenberg and

Moore [1] showed that this adjoint pair gives rise to a triple -1\ lT = (T, 11 ,lJ) in the category!. Let! denote the category of -II lT-algebras. There is a canonical functor ~: ! -+-! • The

adjoint pair F --t U is ca1led tripleable if ~ is an equivalence

of categories. If F' and Fare two left adjoints of U, then

the canonical isomorphism F = F' will give rise to an isomorphism

1 of triples TT = TT , hence an isomorphism of the algebra - -' categories !II =!II. Then ~ is an equivalence iff ~'is. Thus

a functor U : B -+- A is called tripleable if U has a left adjoint d o F and the adjoint pair F ---t U is tripleable. A pair X ==::: y dl of maps in ! ls called U-split if we have a so called

split coequalizer diagram in A:

Ud 0 ~ UX UY Z Udl -s t -78-

with the properties

(i) d(Ud ) d(Ud ) (U) o = l ds = IZ (iii) (Udl)t = sd (iv) (Udo)t = ~ .

A split pair is an i~-split pair. ! has coegualizers of U-split pairs if each U-split pair of maps in ! has a coequa1- izer in B. U preserves coegualizers of U-split pairs if ~ f whenever X ====: y is U-sp1it and has a coequa1izer Y ----+ W dl in B, then Uf = coeq (Ud ,Ud ). We say that U-ref1ects - 0 l coegua1izers of U-sp1it pairs if X ::::: y ----+ W being mapped into a split coequalizer diagram by U imp1ies that X ----+ Y ----+ W ----+ is a coequa1izer diagram in!. Beck [1] & [2] proved the following:

Theorem A. F--t U : ! - !! is tripleab1e if and only if it satisfies the fo11owing conditions:

PTT (1) ! has coequa1izers of U-sp1it pairs

PTT(2) U preserves coequa1izers of U-split pairs

PTT (3) U ref1ects coequalizers of U-sp 11 t pai rs

or PTT(3)': U ref1ects isomorphisms.

Corol1ary. F~U: B - A is tripleab1e if it satisfies the fol1owing conditions: -79-

CTT(1) ! has coequalizers

CTT(2) U preserves coequalizers t CTT(3): U reflects coequalizers or CTT(3)': U reflects isomorphisms.

Theorem B. F--I U : ! - !. is tripleable if it satisfies VTT(l) : Every U-split pair of maps in! splits, and also one of PTT (3) , PTT(3)'.

A functor U : ! - ! is called PTT, CTT or VTT if it satisfies the conditions of Theorem A, Corollary of Theorem A, or Theorem B respectively.

Proposition 5.1. In the following commutative diagram of categories and functors,

D B

A

if (i) U is tripleable

U satisfies PIr(3) (H) 2 (Hi) U has a left adjoint l -80-

then U is tripleable. I

Proof. We will show that U is PTT. Sinee every I

pair of maps in ~ is U-split, it follows that U satisfies UI-split I d PTT(I) • Let X~Ybe a UI-split pair of maps in D. Suppose -dl d o d x y z

is a eoequalizer diagram in~; then Ud eoeq (Ud ' Ud ) , sinee = o l U preserves eoequalizers of U-split pairs.

i8 U split, and sinee U refleets eoequalizers of U -split pairs 2 2 2 we have

Uld - eoeq (Uldo' Uldl ) which proves that U satisfies PTT(2). Pinally, U satisfies I PTT (3) , and so does Ut. -81-

Corollary: Assume U2 : !!. - ! is trip1eab1e.

Then U : D ~ B is trip1eab1e if U U is trip1eab1e and U 1 2 1 1 has a 1eft adjoint.

The converse of the above coro11ary is fa1se, i.e.

the composition of trip1eab1e functors is not necessari1y

trip1eab1e. For examp1e, take ~ = category of torsion free abe1ian groups and homomorphisms; !!. = category of abe1ian groups

and homomorphism; ! ~ category of sets and mappings, together with the forgetfu1 functors.

Proposition 5.2.

is trip1eab1e if one of the fo11owing conditions is satisfied:

(a) U is CTT and U is trip1eab1e 1 2 (b) U is trip1eab1e and U is VTT. 1 2 Proof. Let U .. U U , then U has a 1eft adjoint if 2 1 U and U have. 1 2

(a) We will prove that U • U2U1 satisfies PTT.

Now, D has coequa1izers and in particu1ar ~ has coequa1izers

of U-sp1it pair. Let

d x ====~y z -82-

be a diagram in~, and suppose that (do ,dl) is U-split. If d = coeq (do,dl ), then Uld = coeq (Uldo' Uldl ). Also (Uldo' Uldl ) is U split, and so U U d coeq (U U d ' U U d ), which proves 2 2 l = 2 l o 2 l l that U satisfies PTT(2). If Ud coeq (Ud ' Ud ) , then = o l

Uld = coeq (Uldo' Uldl ) since U2 reflects coequalizers of U2-split pair. If we assume that U satisfies CTT(3), then d coeq (do,d ). 1 = l However, if U satisfies CTT(3)', then by virtue of the fact 1 that U2 satisfies PTT(3)', it is immediate that U satisfies PTT(3)'. This proves that U is trip1eable. do (b) Let X ====t y be U-split. Then (U1do' Uldl ) is dl U -split, and since U satisfies VTT(l) , (do,d ) is U -split. 2 2 l 1

It follows that (do ,dl) has ~ coequalizer. Let

X y z

be a coequa1izer diagram in ~ with (do ,dl) U-split. Then

(Uldo' Uldl ) is U2-sp1it and hence splits by VTT(l) of U2• By

PTT(2) of U1, U1d - coeq (Uldo' Uldl ). U2 satisfies PTT(2) and so Ud • coeq (Ud ' Ud ). This proves that U satisfies PTT(2). o l Clearly U satisfies PTT3 (respectively PTT(3)') if U2 satisfies PTT 3 , (respectively PTT(3)').

Remark. Proposition 5.1 and Proposition 5.2 have -83-

been mentioned by Professor M. Barr in his lectures on triples at McGill University (1969-70).

Proposition 5.3. Let F--4U : .!! - ! be an adjoint pair. If the back adjunction e: : FU - i~ is an isomorphism, then U is VIT. d Proof. Verification of VTT{l): Let X ~ Y be a -dl U split pair of maps in B. Then we have a split coequalizer diagram in !

Ud d o ux UY w s

Ut and hence a split coequalizer diagram in !: FUd o Fd FUX FUt FW Fs

FUt

Moreover, e: : FU - i~ 18 a natural isomo rphism, and 80 we have the following commutative diagrams: -84-

FUd 0 Fd FUX FOY PW FUd l FUt

== ==

d x o y

t where Fa == s' (F\~ 1 FOY 1 Y)

== Fd d' (Y FOY 1 PW) .

Clearly, d d' o X ==::::: Y :;:,::::;::::= PW dl s'

t is a split coequalizer diagram in! and so d' • coeq (do,d ). l

f Verification of PTr(3)': Let X -~-+I Y in ! be such that Uf : UX ---+. uy is an isomorphism. Then FUf : FUX - PUY is an isomorphism. By naturality of FU ---+'l~, we ·have the following commutative diagram -85-

x f y

::

FUX ------+1 FUY FUf which implies that f is an isomorphism.

Corollary. If B is a full reflective subcategory of

!' then the inclusion functor U : ! ---+! is VTT.

Proof: Let F ---t U : ! -!. Then it is well-known that E: FU ----+ id i8 an isomorphism. The result then follows B from the above Proposition.

Theorem 5.4. Consider the following commutative diagram of categories and functors :

D \/ A

Assume that U bas a left adjoint F vith the property 2 2 -86-

that the back adjunction E : F2U2 ----+ i~ is an isomorphism. Then U is tripleable if and only if U is tripleable. I Proof. The necessity part is a combination of Proposition

5.3 and Proposition 5.2. For the sufficiency, we only need to verify that U has a left adjoint, for the rest follows from I

Proposition 5.1. Let F---tU. Then for B € !, D € !!"

Corollary. Let U : ! ----+ ! be the inclusion functor 2

: ~ ----+ of a full reflective subcategory ! of !' then UI ! is

tripleable if and only if U2UI : ~ ----+ ! is tripleable.

Theorem 5.5. The underlying set functor U PC ---+ S

is tripleable.

Proof. .By Proposition 2.2, U has a left adjoint. We -- d o shall show that U satisfies PTT. Let X ==:::: y be a U-split pair dl of maps in PC. Then we have a split coequalizer diagram in S:

d d o X ===;=:: y ~====~. Z • dl s

t -87-

Let

then

OP We assert that R 0 R is the equiva1ence relation generated by R.

It suffices to show that

The necessity fo110ws from the fact that dd dd • For the o = 1 sufficiency, let (Y1'Y2) € Y x Y with dY1 = dY2' Then

d1tY1 = sdY1 = sdY2 = d1tY 2 ; dotY1 = Y1 ; dotY 2 = Y2 •

It then fo11ows that

and

OP OP So we have (Y ,Y ) € R 0 R • It i8 ea8ily seen that R 0 R 1 2 i8 in fact a congruence relation on Y. In order for OP Y/(R 0 R ) to be a pro-f-group, it remains to show that OP R 0 R i8 closed. Let

where p • y x y x y ~ y x y are the i -88-

continuous map defined by Pi(Yl'Y2'Y3) = (Yi ,Y3); i = 1,2. R is closed, being the image of a compact set in a compact

Hausdorff space under the continuous map (do,d ): X --+ Y x Y. l OP Hence D is closed. But R 0 R is the image of D under the closed map P3: Y x Y x Y OP OP and so R 0 R is closed. Hence Y/(R 0 R ) is a pro-f.-group.

We shall show that d o 1T X y

is a coequalizer diagram in PC, where 1T is the quotient map. d' If Y - W in PC has the property d'do = d'dl' then the map

defined by

1/1 [y] .. d'y

is a well-defined morphism in PC and is unique with the property

1/I 0 1T - d'. This proves that U satisfies PTT(l).

Note that the map d: Y ~ Z induces an

isomorphism = Z

in!, it is then easily seen that U satifies PTT(2) and PTT(3).

: ':~"7""~":"'.' -89-

\

Theorem 5.6. The forgetful functors from PC to the

ca tegories Top, ·PTop, Q. are tripleable.

Proof. In the following, all functors in the diagrams

are forgetful. lt is well-known that the forgetful functor from

CHTop to! is trip1eab1e (see Linton Il)), and so by app1ying coro11ary of Proposition 5.1 to the fo11owing commutative diagram:

PC ) CHTop

~/s

it follows that the forgetfu1 functor U : PC - CHTop i8

trip1eab1e. (U: PC - CHTop has a left adjoint by Proposition 2.2.)

CHTop is a full reflective subcategory of Top. App1y the

Coro11ary of Theorem 5.4 to the fo11owing commutative diagram:

PC 1 CHTop \/

It fo11ows that the forgetfu1 functor U PC - Top is trip1eab1e •

.. :·.:.ï.:Fo/~· ~' .. ~:' " -90-

\ ,

To show that the forgetfu1 functor U : PC - PTop·

is trip1eab1e, we consider the fo11owing commutative diagram:

PC 1 PTop \/

Observe that the forgetfu1 functor PTop - Top satisfies PTT(J).

Thus by Proposition 5.1, P-4 U : PC - PTop is trip1eable.

It is we11-known that the forgetful functor Q ----+ ~ is trip1eab1e. By app1ying the Coro11ary of Proposition 5.1 to the fo11owing commutative diagram:

it fo11ows that the inclusion functor PC ----+ Q is trip1eab1e.

Let U0 : f - ~ be the under1ying set functor. This

induces a theory Tu in S. Let (TU :~) denote its category of a1gebras o o (see Linton [1]). -91-

Theorem 5. 7.

Proof. Let 1T be the triple arising from the adj oint sTf_ .§. pair F--fU : !f-.§.. This induces an adjoint pair F1T~uTf:

(see Eilenberg & Moore [1]). It is we11-known that (TuTf: .§.) :.§.1T

[1]). By Theorem 5.5, !1T : pc. Thus to show PC: (TU: .§.), (see Linton o that one it is enough to show TU ~ T JI ' this amounts to showing o U n UFn ~ n. t. (U , U ), which is a natural transformation has an isomorphism o 0 of functors from ~ to !' n e 1!1. We shall see that

n ~f(x) ----+ n.t. (U , U ) is well-defined by S(x)(G)(f) = S : Fn o 0 where x e Fn, G e l.f l ,f e U~G, and 1jIf : Fu ----+ G is the map G. Let a : G - H be given in PC. induced by f : n - U o Then 01jl )(x) , (aoS(x)(G»(f) D a(S(x)(G) (f» = a~f(x) - (a f and n n ( H)oQ )(f) - S(x)(H)(a f) - 1jI (x) - ~ f(x) • (S(x) anf ao wh1ch By the uniqueness property of 1jIa of ' we have aO~f - 1jIaof ' S(x) 1s a proves that (aoS(x)(G»(f) - (S(x)(H) 0 an)(f) and so

natural transformation.

normal Let 1jI e n.t.(Un,U ) and let ~. be the family of all o 0 _. e Ifl. subgroups N of the free disc~ete group L(n) with L(n)/N Ij/(L(n)/N): (L(n)/N)n ---, L(n)/N

----_._------92-

n n n Let a € L(n) , then nNoa = nN(a) € (L(n)/N) where nN : L(n) ----+ L(n)/N is the canonical projection. Put

~ = ~(L(n)/N)(nNoa). For Ni ~ Nj in ~, we have a canonical map nj,i : L(n)/N ----+ L(n)/N • We claim that nj,i(x ) = ~i' j i Nj

n By naturality of ~, we have nj,i~(L(n)/Nj) = ~(L(n)/Ni)nj,i • Now

nj,i(~j) = nj,i~(L(n)/Nj)(n~j (a» = ~(L(n)/Ni)n;,i(n~j (a»

... HL(n)/Ni)(n ionN )n(a) = 1jI(L(n)/Ni)n~ (a) j , j i

/1 Thus there exists a unique x € lim 04- (L(n)/N) with n~ = ~ N€~

for all N € ~, where w lim (L(n)/N) ----+ L(n)/N i8 the "N 04- N€~

canonical map. We will show that ~(G)(f) = ~f(x) for ail n € f € U G. The map f : n ----+ G induces a homomorphism G f, o L(n) ----+ G. Let K be the kernel of this map, then we have an

injective homomorphism Sf : L(n)/K ----+ G so that L(n)/K € lf.1 . The following diagram y::( p(n)~ ":.. -t ~,-nK~ ~~ L(n)/K -93-

is commutative, where n is the canonical map induced by IjJ by the n - L(n)/N, (N € ~); af°'lrKon" = f and so Sf°'lr"K = f uniqueness property of IjJf' By the naturality of 1jJ, we have n SfljJ(L(n)/K) = IjJ(G)Sf • Rence

This proves that 6 is surjective.

Suppose now that 6(x) = 6(x'), x ~ x'. There exists

an open normal subgroup N of F(n) such that x' i xN. Let

be the canonical projection, and

n n- F(n)

the canonical inclusion. Note that F(n)/N € Ici. One has

e(X)(F(n)/N)(;Non) - 1jJ __ (x). ;N(X) • 'liNon

But, ;'N(X) ~ ;N(X'), and hence e(x) ~ e(x'), the desired contradiction.

Naturality of e is easily verified. -94-

REFERENCES

Jon Beck

[1] Untit1ed manuscript, Corne11 (1966).

[2] Triples, Algebras, and Cohomology, Dissertation, Columbia University, New York, 1967.

E.Binz, J.Neukirch and G.H.Wenze1

[1] Free pro-f-products, Queen's Mathematical Preprints

No. 1969-17, Queen's University, Kingston, Canada. N.Bourbaki

[1] General Topology, Parts 1 & 2 (English Translation), Addison Wesley, 1966.

I.Bucur and A.De1eanu

[1] Introduction to the theory of Categories and Functors,

Wiley-Interscience Publication Volume XIX, New York, 1968.

A.Douady

[1] Cohomologie des groupes compacts totalement discontinus,

Sem. Bourbaki, No.189, decembre 1959.

James Dugundji [1] Topology, Allyn and Bacon, Inc., Boston, 1966.

S.Eilenberg and J.C.Moore

[1] Adjoint functors and triples, Ill. J. Math. 9,

381-398, (1965). -95-·

K.Gruenberg

[1] Profinite groups, Frohlich: Algebraic Number Theory,

Thompson Book Co., Washington, 116-128, (1967).

Edwin Hewitt

[1] A remark on density characters, Bull. Amer. Math. Soc.

52, 641-643, (1946).

Serge Lang

[1] Rapport sur la Cohomologie des Groupes, W.A.Benjamin,

Inc., New York, 1966.

Michael Lazard

[1] Groupes analytiques p-adiques, Publications Math.

de l'I.R.E.S. No. 26, 1965.

F.E.J.Linton [1] Some aspects of equationa1 categories, Proc. Conf.

Categ. A1g. (La Jolla, 1965), Springer, Berlin,

84-94, (1966).

W.Magnus, A.Karrass, and D.Solitar [1] Combinatoria1 Group Theory, Interscience, New York, 1966.

E.Manes

[1] A triple theoretic construction of compact algebras,

Seminar on Triples and Categorical Romology Tbeory,

Springer-Ver1ag, Berlin, 91-118, (1969). -96-

\

B.Mitchell

[1] Theory of Categories, Academic Press, New York, 1965.

D.Montgomery and L.Zippin

[1] Topo1ogica1 Transformation Groups, Interscience,

New York, 1955.

L.Ribes

[1] Introduction of Profinite Groups ~nd Galois C~homo1ogy,

Queen's paper in Pure and App1ied Mathematics No.24,

Queen's University, Kingston, Canada, 1970.

E.Schenlanan

[1] Group Theory, Van Nostrand, New York, 1965.

W.R.Scott [1] Group Theory, Prentice-Ha11, Inc., New Jersey, 1964.

J.P.Serre [1] Cohomologie Galoisienne, Springer-Ver1ag, Berlin, 1965.

W.Hurewicz & H.Wa11man

[1] Dimension Theory, Princeton, 1941.