On the Local Theory of Profinite Groups

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12-2020

Mohammad Shatnawi Western Michigan University, [email protected]

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This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. ON THE LOCAL THEORY OF PROFINITE GROUPS

Mohammad Shatnawi

A dissertation submitted to the Graduate College in partial fulfilment of the requirements for the degree of doctor of Philosophy Mathematics Western Michigan University December 2020

Doctoral Committee:

Clifton Edgar Ealy Jr, PhD., Chair John Martino, PhD. Jeffry Strom, PhD. Justin Lynd, PhD. ON THE LOCAL THEORY OF PROFINITE GROUPS

Mohammad Shatnawi, Ph.D.

Western Michigan University, 2020

Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H0 an abelian group where H is a subgroup of G and

H0 is the derived group of H. One important ﬁrst application is Burnside’s normal p- complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it.

Burnside Theorem. Let G be a ﬁnite group, and let P be a sylow p-subgroup that is contained in the center of its normalizer, then G has a normal subgroup H which has elements of P as its coset representatives.

Emil Artin in 1929 [1] Extended the definition to the situation G is infinite and H is a subgroup of G of finite index. The first place the transfer homomorphism appeared in a textbook was in 1937, Lehrbuch der Gruppentheorie, by Hans Zassenhaus [28]. In 1959 the transfer was popularized in the mathematical community in America by Marshall Hall in the textbook The Theory of Groups [14] where the transfer ”in German Die Verlagerung” as a homomorphism from a group G into a subgroup H of finite index was introduced. In an effort to define the transfer homomorphism for profinite groups, Oliver Schirokauer in 1996 [21] published a paper in which he presented a new definition for the standard cohomological transfer as an integral. In this study we will give a definition of the transfer homomorphism for profinite groups which is an analog of Marshall Hall’s definition of the transfer homomorphism for finite groups. Our definition depends on the group action and structure, in relation to axiom of choice and ordinal numbers, using the permanent map. © 2020 Mohammad Shatnawi ACKNOWLEDGMENTS

Firstly, I would like to express my sincere gratitude to my advisor Prof. Clifton Edgar Ealy, Jr for the continuous support of my Ph.D study, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. I’m very grateful to my father and mother Ali and Rifqah Shatnawi, my sisters Rayah, Nusaibah, Sana’, Areej, Azhar, Fatimah, and Bara’ah, my brothers Abraham and Abed ElRahman for their endless support and love without which I won’t be able to reach this point. I’m very grateful to have my wife Tasnim Alhaddadein, who shared my journey and supported me with all her strength to fulﬁll our dream. I’m grateful to have Elias, Ryan, Farah and Jawan in my life who supported me in their own way. I would like to thank the committee members for taking the time in reading my dissertation. I would like to express my gratitude to Dr. Kirsty Eisenhart, thank you for being my sounding board and help. Finally, I would like to thank the Mathematics department faculty, staff and graduate students at Western Michigan University for the unique experience and the wonderful journey which I will never forget.

Mohammad Shatnawi

ii TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

1 Statement of Main Results 1

2 Introduction 4

2.1 Finite Groups ...... 4

2.2 Topological Groups ...... 10

2.3 Profinite Groups ...... 12

2.4 Sylow Theory for Profinite Groups ...... 18

3 Pro-p Groups 22

3.1 Preliminaries ...... 22

3.2 Example: The Profinite Dihedral 2-Group ...... 23

3.3 Example: The Profinite Generalized Quaternion Group ...... 27

3.4 Profinite Hamiltonian Groups ...... 31

3.5 Sylow Intersections in Profinite Groups ...... 33

4 The Transfer Homomorphism for Profinite Groups 36

4.1 Necessary Background in Set Theory ...... 36

4.2 The Permanent Mapping ...... 38

4.3 The Transfer of Profinite Groups ...... 40

4.4 Applications of the Profinite Transfer ...... 48 TABLE OF CONTENTS - Continued

5 Future Work 54

Bibliography 56

Chapter 1

Statement of Main Results

In Chapter 3, we give some examples of pro-2 groups like the proﬁnite dihedral group:

Example 1.0.1. The proﬁnite dihedral group D∗ is the inverse limit of the inverse system of ﬁnite dihedral groups, D , n N. n ∈

Example 1.0.2. The inverse limit of the dihedral 2-groups D2n is isomorphic to D2∞ . and we give an extension of the generalized quaternion group as a proﬁnite group:

Example 1.0.3. The proﬁnite generalized quaternion group Q∗ is the inverse limit of ﬁnite generalized quaternion groups.

We characterize the proﬁnite Hamiltonian groups as an inverse limit of ﬁnite Hamiltonian groups.

Theorem 1.0.4. Let H be a pro-Hamiltonian group. Then H is a profinite abelian group, or H = Q E2 A where E2 is an elementary abelian pro-2 group, Q is ∗ × ∗ × ∗ ∗ ∗ the profinite generalized quaternion group and A∗ is a profinite abelian group in which every element is of odd Steintz prime order.

We also extend some results of sylow intersections to proﬁnite groups:

Lemma 1.0.5. Let G be a proﬁnite group, and let K = (S S Syl (G)). Then: | ∈ p T 1 (a)K E G, and K is a pro-p subgroup of G.

(b) K contains every normal pro-p subgroup of G.

Theorem 1.0.6. Let G be a proﬁnite group, let D = P Q be a maximal sylow p- ∩ intersection of P,Q Sylp(G) with P , Q. Then: ∈

(a) D is a proper subgroup of each sylow p-subgroup of NG(D).

(b) Any two distinct sylow p-subgroups of NG(D) have intersection D.

taining D and the sylow p-subgroups of NG(D).

The main result of this study is in chapter 4 in which we introduce the transfer for profinite groups using the permanent map. Let G be a profinite group, H be a closed subgroup of G of not necessarily finite index, and K be a normal subgroup of H such that H/K is abelian. Let Y be a transversal of H in G and Y = n SN, where SN is | | ∈ the set of all Steinitz numbers. For an infinite Steinitz number n we say that a monomial (n n)-matrix A over a commutative ring with identity R is known if the limit ×

lim ∏ am i n m i → ≤ exists and converges in R regardless of the order of the indices, where a R is the m ∈ nonzero entry in the m-th row of A.

Let M (H/K) be the group of all (n n) known monomial matrices with entries n × from H/K. We deﬁne the transfer homomorphism of proﬁnite groups as a composition of the monomial representation

µ : G M (H/K) → n

2 and the permanent map perm : M (H/K) H/K. n n →

This definition is an analog for the one of the transfer for finite groups in [14]. In an effort to define the transfer homomorphism for profinite groups, Oliver Schirokauer in 1996 [21] published a paper in which he presented a new definition for the standard cohomological transfer as an integral, our aim is to define the transfer using the group structure and the permanent map. In [10] the authors proved Burnside Normal p-Complement Theorem for Profinite Groups

Theorem 1.0.7 (Burnside Normal p-Complement Theorem for Proﬁnite Groups). Let G be a proﬁnite group, and let P be a sylow pro-p subgroup that is contained in the center of its normalizer, then G has a closed normal subgroup H which has elements of P as its coset representatives.

without using the transfer, instead they use the following lemma

Lemma 1.0.8 ([10]). Let G be an inverse limit of ﬁnite groups Gi where i is in a directed set I. Then G has a normal sylow p-complement if and only if Gi has a normal sylow p-complement for all i I. ∈

We give a proof of this theorem using the transfer following the work in [14] chapter 14, also we used the transfer to prove

Theorem 1.0.9 (Focal Subgroup Theorem for Proﬁnite Groups). Let G be a proﬁnite group, P a pro-p sylow subgroup of G and G0 be the derived group of G. Then

G V (G) P/P G0 P ∩ which was also been proved in [10] for proﬁnite groups by extending from the ﬁnite version without using the transfer.

3 Chapter 2

Introduction

Profinite groups arise naturally in mathematics as finite groups, Galois groups, the p- adic integers, p P, G/N where G is any compact Hausdorff group and N is the con- ∈ nected component of the identity of G. Profinite groups have a Sylow theory. The Local Theory of finite groups is based on the Sylow theory of finite groups. It asks what global properties of a group G can be deduced from the knowledge of P, a Sylow p-subgroup of G, and NG(P). The collection of all subgroups of G, H, such that H = NG(Q) where Q is a p-subgroup of G for some prime p which divides the order of G are called the p-local subgroups of G. Burnside’s p-complement theorem, as an application of the Transfer homomorphism, is an example of such a result. This may also be viewed as a first application of fusion in finite groups. This chapter will summarize the necessary background for this dissertation.

2.1 Finite Groups

In this section we present the basic theory which is important to our intent to extend results related to ﬁnite groups to proﬁnite groups. Following chapter 14 in Marshal Hall [14] for the convenience of the reader we give the following.

Deﬁnition 2.1.1. Let G be a group, and H G of ﬁnite index, take S = u , ,u , be ≤ { 1 ··· n}

4 the set of left coset representatives of H in G. A monomial permutation M is a mapping u h u where i = 1, ,n, and j = i( j). i → i j j ··· Remark 2.1.2. If M is the set of all monomial permutations, we can make it into a group via the multiplication: for M ,M M, with M : u h u and M : u h u , we 1 2 ∈ 1 i → i j j 2 j → jk k write M M : u (h h )u , and the identity of M is u u . 1 2 i → i j jk k i → i Remark 2.1.3. let R be a monomial representation of a ﬁnite group G with multipliers from H G. set ≤ π(g) : u h u i, j = 1,....,n i → i j j then the mapping

g hi j H0 n n → × is a homomorphism of G into Mn(H/H0).

Definition 2.1.4. Let G be a finite group, H G, and let S = u , ,u to be the ≤ { 1 ··· n} set of left coset representatives of H in G. Set φ(z) = x j for z = hx j. The transfer homomorphism is defined from G into H/H0 via

n 1 VG H(G) = ∏xigφ(xig)− mod H0 → i=1

Remark 2.1.5. We note that the map φ can be considered as a choice function, which we will call ν in chapter 4.

Theorem 2.1.6 ([14]). (a) The mapping g VG K(g) is a homomorphism of G into → → K/K0.

(b) The transfer VG K is well deﬁned; i.e independent of the choice of representa- → tives.

(c) if T K G, then VG T (g) = VK T [VG K(g)] . ≤ ≤ → → → The next result is known as Burnside normal p-complement theorem for ﬁnite groups.

5 Theorem 2.1.7 (Burnside’s Theorem [14]). Let G be a ﬁnite group, and let P be a sylow p-subgroup that is contained in the center of its normalizer, then G has a normal subgroup H which has elements of P as its coset representatives.

The following lemma is used to show the above theorem.

Lemma 2.1.8. Let G be a ﬁnite group, if two subsets K1 and K2 of G are normal in a sylow p-subgroup P of G and are conjugate in G then K1 and K2 are conjugate in

NG(P).

This leads to a well known results for ﬁnite groups, such as theorems of P. Hall, Grun,¨ and Wielandt. The importance of these results arise from the relationship between the sylow p-subgroup of a group G and the factor group G/K of G which are p-groups. Now, Let H H H G such that G : H = n so that 0 ≤ 0 ≤ ≤ | |

n G = Hxi i[=1 and n 1 VG H(g) = ∏xig[φ(xig)]− H0. → i=1

1 For g G and i = 1,2, ,n we deﬁne ig 1,2, ,n such that x gx− ∈ ··· ∈ { ··· } i ig ∈ H. Then for a ﬁxed g G we have the permutation π(g) : i ig of the transitive ∈ → permutation representation of G on the left cosets of H in G. Thus we can write

1 VG H(g) = ∏xigxig− H0 → i in the permutation π(g) we can write it as t disjoint cycles

t π(g) = ∏ σ j j=1 choose the least element in each σ and form the set C (g). For i C (g) let r = σ j H ∈ H i | j|

6 where i appears in σ j. Then

∑ ri = n i C (g) ∈ H Lemma 2.1.9 ([14]).

ri 1 VG H(g) = xig xi− H0 → ∏ i C (g) ∈ H

ri 1 1 Where xig xi− is the ﬁrst power of xigxi− which lies in H.

The following is needed for the proof of P. Hall theorem and Hall-Wielandt theorem in [14].

Deﬁnition 2.1.10. The diagonal contribution, d(g) is the contribution of cycles of length one in π(g) to VG H(g) is given by →

1 d(g) = ∏ xigxi− H0 i=ig

n Remark 2.1.11. Note that d(g) = ∏i=1 hii with π(g) = [hi j H0]n n. ×

1 Lemma 2.1.12 ([14]). If u and v are conjugate in G, then d(u) = d(v), and d(u− ) = 1 [d(u)]− .

Now, for h H we deﬁne d (h) h 1d(h). Using lemma 2.1.12, we have ∈ ∗ ≡ −

Lemma 2.1.13 ([14]). If h H, then ∈

n ri ri 1 VG H(h) h d∗(h )d∗(xih− xi− ) H0 → ∏ ≡ i C (h) ∈ H

Corollary 2.1.14. If d∗(h) H0 for all h H, then for any h H we have VG H(h) = ∈ ∈ ∈ → hn.

We set up the stage for the main theorems in this section from chapter 14 in [14]. Let p P, and G any ﬁnite group, deﬁne ∈

O p(G) = x G ( x , p) = 1 h ∈ | | | i 7 thus G/O p(G) is the maximal p-factor group of G. Let P be a sylow p-subgroup of G, and deﬁne N = NG(P), and let H be any subgroup of G containing N. Now set P∗ = P O p(G), N = N O p(G), and H = H O p(G), so that G = O p(G)P = O p(G)N = ∩ ∗ ∩ ∗ ∩ O p(G)H and P/P = N/N = H/H = G/O p(G) . Here O p(G) is fully invariant | ∗| | ∗| | ∗| | | p subgroup of G, note that P∗ is a sylow p-subgroup of O (G), and N∗ normalizes both O p(G) and P, and so N normalizes O p(G) P = P . Now O p(O p(G)) = O p(G) but ∗ ∩ ∗ p p is may happen that O (H∗) $ H∗. If We consider the case when O (H∗) $ H∗. Note p that O (H∗) is fully invariant subgroup of H∗, and H∗ is normal in H. Since H/H∗ is a p p p-group we have O (H) = O (H1). Now we deﬁne

p p H0 = H∗ [H∗,H]O (H∗) where

p p H∗ = x x H∗ h | ∈ i and

[H∗,H] = [h∗,h] h∗ H∗, h H h | ∈ ∈ i

p p Since H/O (H∗) is a ﬁnite p-group and so it is also nilpotent. This means that [H∗,H]/O (H∗) H /O p(H ). Now, by deﬁnition, H p T where T H with H : T = p such that ∗ ∗ ∗ ⊂ ∗ | ∗ | p p [H∗,H]O (H∗) $ T $ H∗. So, if O (H∗) $ H∗ we have H0 C H∗ and H∗/H0 is a p- group.

Theorem 2.1.15 (P. Hall [14]). Let G be any ﬁnite group, P a sylow p-subgroup, N = p NG(P), and H a subgroup containing N. Let O (G) be deﬁned as before, and set P∗ = P O p(G),N = N O p(G), and H = H O p(G). Then O p(H ) = O p(H), and if ∩ ∗ ∩ ∗ ∩ ∗ p p p O (H∗) , H∗, then H0 = H∗ [H∗,H]O (H∗) is a proper subgroup of H∗, and H∗ can be obtained by adjoining to H0 certain conjugates lying in H∗ of elements

(p-1) e (u,z ) = [u,z , ,z ] p i i ··· i z }| {

8 where u P and z P for i = 1, ,s. If ∈ ∗ i ∈ ···

s G = HtiP i[=1 is a decomposition of of G into double cosets of H and P, let π for i = 1, ,s be the i ··· transitive representation of P on the cosets of H in HtiP. Then πi is not of degree one, and we choose zi so that πi(zi) is of order p in the center of π(P).

Corollary 2.1.16 ([14]). If e (u,z) = 1 for all u,z P , then p ∈ 1

O p(N ) = N = O p(G ) N 1 1 ∩ and

p p G1/O (G1) = N1/O (N1)

This will happen in particular if the class of P1 is less than p.

Theorem 2.1.17 (Hall-Wielandt [14]). Let P1 be a sylow p-subgroup of G1 and Q1 be a weakly closed subgroup of P1. Let N1 = NG1 (P1) and H1 = NG1 (Q1). Then any one of the following conditions will ensure O p(H ) = H = O p(G ) H , and so G /O p(G ) = 1 1 ∩ 1 1 1 p H1/O (H1).

1.e (u,z) = 1 for all u P and all z Q . p ∈ 1 ∈ 1

2.e (u,z) = 1 for all u,z Q . p ∈ 1

3.Q 1 Zp 1(P1), where Zp 1(P1) is the (p 1)st term of ascending central series ⊆ − − − for P1.

Corollary 2.1.18 ([14]). Let Q1 be a characteristic subgroup of P1. If Q1 is not weakly closed in P1, then there is another sylow p-subgroup P2 which contains Q1 but in which

Q1 is not normal. This must be the case if Q1 satisﬁes either one of the conditions in p p Hall-Wielandt theorem, but G1/O (G1) H1/O (H1).

9 Theorem 2.1.19 ([14]). Let P be a sylow p-subgroup of G. Then VG P(G) P/(P G0). → ∩

Theorem 2.1.20 (First Theorem of Grun¨ [14]). Let P be a sylow p-subgroup of G. Then

VG P(G) P/P∗, where →

1 P∗ = [P N (P)] (P z− P0z). ∩ G0 ∩ z G [∈

Theorem 2.1.21 (Second Theorem of Grun¨ [14]). If G is p-normal, then

p p G/O (G)G0 N(Z(P))/O (N(Z(P)))(N(Z(P)))0 where P Syl (G). ∈ p

Theorem 2.1.22 ([14]). [Hall-Grun]¨ If G is p-normal, then the greatest factor group of G which is a p-group is isomorphic to that for the normalizer of the center of a sylow p-subgroup.

Our aim is to give proves of the above for proﬁnite groups.

2.2 Topological Groups

In this section we follow [25].

Deﬁnition 2.2.1. A group G is said to be a topological group if it satisﬁes the following conditions:

1. G is a topological space, and

2. The map φ : G G G, given by φ(x,y) = xy 1 is continuous. × → −

Theorem 2.2.2 ([25]). Let G be a topological group.

10 (a) the map (x,y) xy from G G to G is continuous. and the map x x 1 from G 7→ × 7→ − to G is a homeomorphism. For g G the maps x xg and x gx from G to G ∈ 7→ 7→ are homeomorphisms.

(b) Every right or left coset of an open (closed) subgroup of G is open (closed).

(c) Any subgroup of G is a closed subgroup, and any closed subgroup of G of ﬁnite index is open. Moreover if G compact then any open subgroup is of ﬁnite index.

(d) If H is a subgroup containing a non-empty open subset U of G then H is open in G.

(e) If H is a subgroup of G and K EG then H is a topological group with respect to the subgroup topology and G/K is a topological group with respect to the quotient

topology, and the quotient map q : G G/K takes open sets to open sets. →

Lemma 2.2.3 ([25]). Let G be a compact topological group. If C is a subset which is both closed and open such that 1 C, then C contains an open normal subgroup. ∈

Proposition 2.2.4 ([25]). Let G be a compact totally disconnected topological group.

(a) Every open set in G is a union of cosets of open normal subgroups.

(b) A subset of G is both closed and open if and only if it is a union of ﬁnitely many cosets of open normal subgroups.

X = NX N an open normal subgroup of G ∩{ | }

In particular

C = NC N an open normal subgroup of G ∩{ | }

11 for each closed subset C, and the intersection of the open normal subgroups of G is the trivial subgroup.

Proposition 2.2.5 ([25]). Let G be a compact totally disconnected topological group.

(a) Every open set in G is a union of cosets of open normal subgroups.

(b) A subset of G is both closed and open if and only if it is a union of ﬁnitely many cosets of open normal subgroups.

X = NX N C G { o } \

In particular

C = NC N C G { o } \

for each closed subset C, and NCoG N = idG. T Lemma 2.2.6 ([25]). Let G i I be a family of topological groups, let { i ∈ }

C = Gi i I ∈ We can make C into a topological group by taking the product topology and deﬁning multiplication pointwise.

2.3 Proﬁnite Groups

In this section we follow [25].

Deﬁnition 2.3.1. A directed set is a partially ordered set (or a poset), I, such that for all i ,i I there is a j I with i j and i j. 1 2 ∈ ∈ 1 ≤ 2 ≤

12 Deﬁnition 2.3.2. An inverse System (Gi,φi j,I) consists of ﬁnite groups Gi with continuous homomorphisms φ : G G with the property: Whenever i, j,k I such that i j j → i ∈ i j k we have ψ = id and φ φ = φ . Where I is a directed poset. ≤ ≤ ii Gi i j jk ik

Example 2.3.3. Let I = N, and consider the system (Z/piZ,φ ,I), and for i j in I i j ≤ deﬁne φ : Z/p jZ Z/piZ. Indeed (Z/piZ,φ ,I) is an inverse system. i j → i j

Deﬁnition 2.3.4. Let (Gi,φi j,I) be an inverse system of ﬁnite groups and let Y be a group, the family (ψi : Y Gi)i I of continuous maps is called compatible if φi jψ j = ψi → ∈ whenever i j in I. ≤

Definition 2.3.5. An inverse limit (G,φi,I) of an inverse system (Gi,φi j,I) of finite groups is the profinite group G with compatible continuous homomorphisms φ : G G i → i with the property: Whenever (ψi : Y Gi)i I is a compatible family of continuous → ∈ homomorphisms from a group Y there is a unique homomorphism ψ : Y G such that → φ ψ = ψ for all i I. i i ∈

Example 2.3.6. The inverse limit of of the inverse system in example 2.3.3 is the p-adic integers Zp.

Definitionb 2.3.7. The inverse limit of an inverse system (Gi,φi j,I) of finite groups is called a profinite group G denoted by

G = limGi i I ←−∈

The following proposition gives a description of proﬁnite groups.

Proposition 2.3.8 ([25]). Let (Gi,φi j,I) be an inverse system

(a) The inverse limit G is unique up to isomorphism.

(b) If C = G i I and for each i let π be the projection map from C to G . { i ∈ } i i

Deﬁne X = c C φ π (c) = π (c) for all i, j with j i { ∈ i j i j ≥ }

13 and ψ = π for each i. Then (X,ψ ) is an inverse limit of (G ,φ ,I) i i|X i i i j

then X is called the special inverse limit group denoted by slimGi, and the maps ←− ψi are continuous homomorphisms.

Proposition 2.3.9 ([25]). Let (Gi,φi j,I) be an inverse system, and set G = limGi the ←− proﬁnite group. Then

(a) If each Gi is Hausdorff, so is G.

(b) If each Gi is totally disconnected, so is G.

C = Gi i I ∈

(d) If each Gi is compact and Hausdorff, so is G.

(e) If each Gi is non-empty compact and Hausdorff group, then G is non-empty.

Proposition 2.3.10 ([25]). Let (G,ψi) be the inverse limit of the inverse system (Gi,φi j,I) of non-empty compact Hausdorff groups. Then

(a) ψi(G) = j i φi j(G j) for each i I. ≥ ∈ T 1 (b) The set ψ− (U) with i I and U open in G form a base for the topology on G. i ∈ i

(d) If θ is a map from a group Y to G, then θ is continuous if and only if each map

ψiθ is continuous.

(e) If f : G A is a continuous map to a discrete space, then f factors through G for → i some i I; i.e for some i there is a continuous map g : G A such that f = gφ . ∈ i → i

14 The next proposition gives an insight on the characterization of proﬁnite groups.

Proposition 2.3.11 ([25]). Let G be compact Hausdorff totally disconnected group, then G is the inverse limit of its discrete quotient groups.

We introduce some notations and disclaimers: for a proﬁnite group G we will write H G to mean that H is closed subgroup of G, and when it is open we will write ≤ H G. ≤o

Deﬁnition 2.3.12. A ﬁlter base in G is a family I of open normal subgroups of G such that for all K ,K I there is a subgroup K I with K K K . 1 2 ∈ 3 ∈ 3 ⊂ 1 ∩ 2

The next two results set up the stage for the characterization of proﬁnite groups.

Proposition 2.3.13 ([25]). Let (G,ψi) be an inverse limit of an inverse system (Gi,φi j,I) of compact Hausdorff topological groups and let L C G. Then kerψ L for some o i ≤ i. Consequently G/L is isomorphic as a topological group to a quotient group of a subgroup of some Gi, and if each map ψi is onto then G/L is isomorphic to a quotient group of some Gi.

Proposition 2.3.14 ([25]). Let G be a topological group and I be a filter base of closed normal subgroups, and for K,L I define K L if and only if L K. Thus I is a ∈ ≤0 ≤ directed set with respect to and the surjective homomorphism q : G/L G/K ≤0 KL → defined for K L, make the groups G/K into an inverse system. Let (G,ψ ) = limG/K, ≤0 K ←− there is a continuous homomorphism θ : G G with kernel K I K, with image a dense → ∈ b subgroup of G, and such that ψ θ is the quotient map from GT to G/K for each K I. If K b ∈ G is compact then θ is onto; if G is compact and K I K = 1 then θ is an isomorphism b ∈ of topological groups. T

The term class of ﬁnite groups is a class in the usual sense which also closed with respect to isomorphic images; i.e if C is a class and if F1 C and F1 F2 then ∈ F C . Let C be a class of ﬁnite groups, a group F in C is called a C -group. If G 2 ∈ is the inverse limit of C -groups then we call G a pro-C group. Note that C -groups are

15 pro-C groups. We say that C is closed for subgroups if every subgroup of a C -group is a C -group, same terminology is used for quotient groups. In addition, we say that C is closed for direct products if F F C whenever F C and F C . 1 × 2 ∈ 1 ∈ 2 ∈ One of the important classes are the class of all finite groups, the class of finite p-groups with p P is fixed, and the class of all finite cyclic groups. ∈

Definition 2.3.15. A pro-p group is the inverse limit of finite p-groups, and a procyclic group is the inverse limit of finite cyclic groups.

Theorem 2.3.16 ([25]). Let C be a class of ﬁnite groups, which is closed for subgroups and direct products, and let G be a proﬁnite group. The following are equivalent:

(a) G is a pro-C group.

(b) G is isomorphic as a topological group to a closed subgroup of a Cartesian prod-

uct of C -groups.

(d) G is compact and totally disconnected, and for L Co G there is a subgroup N Co G with N L and G/N C . If in addition C is closed for quotients then (d) can be ≤ ∈ replaced by

(e) G is compact and totally disconnected, and G/L C for every L C G. ∈ o

Taking C = C f in the above theorem we obtain an important characterization of proﬁnite groups:

Corollary 2.3.17 ([25]). Let G be a proﬁnite group. The following are equivalent:

(a) G is a proﬁnite group.

(b) G is isomorphic as a topological group to a closed subgroup of a Cartesian product of ﬁnite groups.

Then next important result gives a description of how proﬁnite groups its subgroups and its factor groups can be viewed as inverse limits.

Theorem 2.3.18 ([25]). Let G be a proﬁnite group. If I is a ﬁlter base of closed normal subgroups of G such that N I N = 1 then: ∈ T

G limG/N N I ←−∈ moreover

H limH/(H N) N I ∩ ←−∈ for each closed subgroup H and

G/K limG/KN N I ←−∈ for each closed normal subgroup K.

The above result also shows that subgroups and factor groups of proﬁnite groups are proﬁnite.

Remark 2.3.19.

1. In the context of proﬁnite groups, any subgroup is closed in G, we will write ” ” ≤ to denote closed subgroups, and ” ” for open subgroups. ≤o

2. If G is proﬁnite, then

G lim G/N N←−/oG

3. If G is proﬁnite and K G, then ≤

K lim K/(K N) lim KN/N ∩ N←−/oG N←−/oG

17 2.4 Sylow Theory For Proﬁnite Groups

In this section we follow Ribes [20] and Wilson [25]. The following are essential for this section.

Proposition 2.4.1 ([20]). Let C ,C , be a countably inﬁnite set of nonempty closed 1 2 ··· subsets of a proﬁnite group G, such that each Ci have empty interior. Then

∞ G , Ci n[=1 and so the cardinality of G is either ﬁnite or uncountable.

Deﬁnition 2.4.2. A Steinitz number, n, is a formal inﬁnite product

n = ∏ pn(p) p P ∈ where n(p) is non-negative natural number or ∞. By SN we mean the set of all Steinitz numbers.

Now we give some properties of Steinitz numbers following Ribes and Zalesskii. [20].

Deﬁnition 2.4.3. Let n(p,i) ni = ∏ p i I ( p P | ∈ ) ∈ be a collection of Steinitz numbers. Then

1. n n if n(p,i) n(p, j) for all p P i j ≤ ∈

n(p) 2. ∏J ni = ∏p P p where n(p) = ∑J n(p,i) with J I. ∈ ⊆

n(p) 3. gcd ni i J = ∏p P p where n(p) = mini n(p,i) i J with J I. { | ∈ } ∈ { | ∈ } ⊆

n(p) 4. lcm ni i J = ∏p P p where n(p) = maxi n(p,i) i J with J I. { | ∈ } ∈ { | ∈ } ⊆

18 With this we can deﬁne the order and the index in proﬁnite groups.

Definition 2.4.4. Let G be a profinite group, and H G. Let U denote the set of all ≤ open normal subgroups of G. The index of H in G is defined as

G : H = lcm G/U : HU/U U U SN | | {| | ∈ } ∈

the order of G, G , is deﬁned as | |

G = G : 1 = lcm G/U U U SN | | | | {| | ∈ } ∈

We proceed with some important properties and extensions from well known classical results.

Theorem 2.4.5 ([20]). Let G be a proﬁnite group.

(a) If H G then G : H is a natural number if and only if H is an open subgroup of ≤ | | G.

(b) If H G then ≤ G : H = lcm G : U H U G | | {| | | ≤ ≤o }

G : H = lcm G/U : HU/U U U 0 | | {| | | ∈ }

(d) If K H G then G : K = G : H H : K . This is Lagrange’s theorem. ≤ ≤ | | | || |

(e) Let H i I be a family of closed subgroups of G ﬁltered from below. Assume { i | ∈ } that H = i I Hi. Then ∈ T G : H = lcm G : H i I | | {| i| | ∈ }

19 (f) Let (Gi,φi j) be a surjective inverse system of proﬁnite groups over a directed poset I. Let G = G . Then limi I i ←− ∈

G = lcm G i I | | {| i| | ∈ }

(g) For any collection G i I of proﬁnite groups, { i | ∈ }

∏Gi = ∏ Gi | i I | i I | | ∈ ∈

With these results in hand, we can begin the sylow theory.

Definition 2.4.6. Let G be a profinite group, and p P. A sylow p-subgroup of G is a ∈ subgroup P such that P possibly infinite power of p and ( G : P , p) = 1. | | | |

The following results are an extension of the classical sylow theorems for ﬁnite groups.

Theorem 2.4.7 ([25]). Let G be a proﬁnite group, and p P ∈

(a) G has a sylow p-subgroup.

(b) If P is a sylow p-subgroup of G and T is a pro-p subgroup of G then T g P for ≤ some g G. ∈

(d) If P and P are sylow p-subgroups of G then Pg = P for some g G. 1 2 1 2 ∈

Theorem 2.4.8 ([20]). Let G be a proﬁnite group, K a normal subgroup and P a sylow p-subgroup of G, p P. Then ∈

(a)K P is a sylow p-subgroup of K. ∩

(b)KP /K is a sylow p-subgroup of G/K.

20 (c)G = NG(Q)K for each sylow p-subgroup Q of K.

(d)H = NG(H) whenever H is a closed subgroup which contains NG(Q) for some sylow p-subgroup Q of K.

21 Chapter 3

Pro-p Groups

3.1 Preliminaries

With section 4 in chapter 2 in mind, we present some known results on pro-p groups.

Theorem 3.1.1 ([20]). A pro-p group is always pronilpotent.

Lemma 3.1.2 ([20]). Let P be a pro-p group, then all maximal subgroups of P are normal.

Lemma 3.1.3 ([20]). Let P be a pro-p group. If H is an open normal proper subgroup of P, then

H < NP(H)

Theorem 3.1.4 ([25]). Let G be a proﬁnite group, and let Z be the proﬁnite completion of Z. b

1. The map δ : Z G G with δ(n,g) = gn for n Z exists and is unique. If g G × → ∈ ∈ and z Z then gz is deﬁned. ∈ b b

2. For g Gb and z ,z Z then gz1+z2 = gz1 gz2 and (gz1 )z2 = gz1z2 . ∈ 1 2 ∈

3. If g ,g G commute andb z Z then (g g )z = gz gz . 1 2 ∈ ∈ 1 2 1 2 b 22 Lemma 3.1.5 ([20]). Any procyclic pro-p group is isomorphic to Z/pnZ for some n 0 ≥ or to Zp. Any procyclic group is isomorphic to a quotient group of Z.

b For the proof of last result see [20]. The next result is a characterizationb of ﬁnite p-groups [14].

Theorem 3.1.6 ([14]). The p-groups of order pn which contain a cyclic subgroup of index p are of the following type:

1. Abelian

(a) If cyclic then it is isomorphic to Z/pnZ.

n 1 (b) For n 2, we have xp − = 1, yp = 1, and xy = yx. ≥

2. Non abelian

n 1 (a) p is odd and n 3 we have xp − = 1, yp = 1, and yx = x1+pn 2y. ≥ − n 1 (b)p = 2, n 3 we have the generalized quaternion group: x2 − = 1, y2 = ≥ 2n 2 1 x − , and yx = x− y.

n 1 (c)p = 2, n 3 we have the dihedral group: x2 − = 1, y2 = 1, and yx = x 1y. ≥ − n 1 n 2 (d)p = 2, n 4 we have : x2 − = 1, y2 = 1, and yx = x1+2 − y. ≥ n 1 n 2 (e)p = 2, n 4 we have : x2 − = 1, y2 = 1, and yx = x 1+2 − y. ≥ −

Note that for a system of abelian cyclic p-group G, we can take the inverse system and have G Zp. The next two sections will provide some generalizations of this results to pro-p groups.b

3.2 Example: The Proﬁnite Dihedral 2-Group

We consider the collection of all ﬁnite dihedral groups, D with n N, represented as 2n ∈

n 2 1 D = r,s r = 1 = s , srs = r− 2n h | i 23 Example 3.2.1. The proﬁnite dihedral group D∗ is the inverse limit of the inverse system of ﬁnite dihedral groups, D , n N. n ∈

Proof. First we identify

n 2 1 D = r,s r = 1 = s , srs = r− 2n h | i

k 2 1 D = r ,s r = 1 = s , s r s = r− 2k h 1 1 | 1 1 1 1 1 1 i

m 2 1 D = r ,s r = 1 = s , s r s = r− 2m h 2 2 | 2 2 2 2 2 1 i and let D∗ be the proﬁnite dihedral group, its generators are given explicitly as

2 2 D∗ = x,y y = (xy) = 1 h | i where x has Steinitz number order. To construct an inverse system of dihedral groups, let I = N be the index set, deﬁning the relation on I by n m if and only if n m, we ≤ ≤ | have I as an indexed set. Deﬁne

φ : D D kn 2n → 2k for any n,k I, such that k n, via φ (ris j) = ri s j, this is well deﬁned since k n and the ∈ | kn 1 1 | exponent on the right are taken modulo k and 2. It is a group homomorphism, to see this let ris j,rlst D and take ∈ 2n

i j l t i l j+t i l j+t φ (r s r s ) = φ (r − s ) = r − s kn · kn 1 1 on the other hand

i j l t i j l t i l j+t φ (r s ) φ (r s ) = r s r s = r − s kn · kn 1 1 · 1 1 1 1 where i j is modulo k and j +t is modulo 2. Consider the system (D ,φ ), we will − 2n kn show that this is an inverse system of ﬁnite dihedral groups. First note that φ : D kk 2k →

24 D , by our deﬁnition this is the identity map. Now, let m k n in I, and consider the 2k | | maps

φ : D D mk 2k → 2m and

φ : D D kn 2n → 2k take φ φ : D D , note that mk ◦ kn 2n → 2m

(φ φ )(ris j) = φ (φ (ris j)) mk ◦ kn mk kn i j = φmk(r1s1)

i j = r2s2

i j i j φmn(r s ) = r2s2

This shows φ φ = φ , and hence the system (D ,φ ) is an inverse system. mk ◦ kn mn 2n kn Consider the proﬁnite dihedral group, D∗, as a subgroup of GL2(Z) as follows: set b 1 z x =   0 1     and 1 z − y =   0 1     where z Z such that z , 0, it is easy to see that x has an inﬁnite Steinitz number order ∈ 1 and y is ofb order 2. Indeed yxy = x− . So

D∗ x,y h i set H = x and K = y , and deﬁne Φ : K Aut(H) by sending k K to the inversion h i h i → ∈ 1 automorphism on H, the associated action is k h = h− . so we have D∗ H o K. From · Φ

25 the semidirect product we have the short exact sequence

1 Z D∗ C2 1 we also have b

1 Z/nZ Dn C2 1 applying the inverse limit functor to the above short exact sequence, we obtain

1 limZ/nZ limDn limC2 1 ←− ←− ←− γ

1 Z D∗ C2 1 by the ﬁve lemma for short exact sequences [9], we deduce that γ is an isomorphism. b

Example 3.2.2. The inverse limit of the dihedral 2-groups D2n is isomorphic to D2∞ .

Proof. We follow the proof of lemma 2.2.1. First note that (D2n ,φmn,I) is an inverse system with I = N and φ is as in lemma 2.2.1 by setting the relation on I as the mn ≤ n usual inequality. We have D2n Z/2 ZoC2, this translates to the short exact sequence:

n 1 Z/2 Z Dn C2 1 now consider D2∞ as subgroup of GL2(Z), using

b 1 z x =   0 1     and 1 z − y =   0 1     where z is a nonzero element in Z, it is easy to see that x has an inﬁnite Steinitz number

1 order and y is of order 2. Indeedbyxy = x− . So

D2∞ x,y h i

26 set H = x and K = y , and deﬁne Φ : K Aut(H) by sending k K to the inversion h i h i → ∈ 1 automorphism on H, the associated action is k h = h− . so we have D2∞ H o K. · Φ From the semidirect product we have the short exact sequence

1 Z2 D2∞ C2 1 then b n 1 limZ/2 Z limD2n limC2 1 ←− ←− ←− λ

1 Z2 D2∞ C2 1 and by the ﬁve lemma for short exact sequences, we deduce that λ is an isomorphism. b

3.3 Example: The Proﬁnite Generalized Quaternion Group

The generalized quaternion group is given by:

2i 4 2i 1 2 y 1 Q = x,y x = y = 1, x − = y , x = x− i h i

i+2 a b It is a non-abelian 2-group of order 2 . Any element in Qi can be written as x y i a b c d where a is modulo 2 and b is modulo 4, and for any x y and x y in Qi the product

b xayb xcyd is xa+( 1) cyb+d, to see this we need to establish the following relation: · −

b b ( 1)b y xy− = x − since b is modulo 2, it it either even or odd; if b is even then

b b y xy− = x (3.1)

i 1 since even powers of y can be absorbed into the powers of x using the relation x2 − = y2. In the case b to be odd, note that any odd number can be written as 2s + 1 for s N, so ∈

27 b 2s y 1 we can rewrite y = y y, and form relation 3.1 and the relation x = x− we have

b b 1 y xy− = x− (3.2) the relations 3.1 and 3.2 together gives

b b ( 1)b y xy− = x − using induction, we can prove that

b c b b b c y x y− = (y xy− ) so

a b c d a b c b b+d a ( 1)bc b+d a+( 1)bc b+d x y x y = x (y x y− )y = x x − y = x − y . ·

Regarded as finite 2-groups we can construct the pro-2 group for a system of generalized quaternion groups. We begin with the construction of the inverse system. To construct an inverse system of generalized quaternion groups, let I = N be the index set, defining the relation on I by n m as the usual inequality, we have I as an indexed ≤ set. Define

φ : Q Q kn n → k for any n,k I, such that k < n, via φ (xiy j) = xiy j, where i is modulo 2n and j is ∈ kn modulo 4. This map is well deﬁned since k < n, it is also a homomorphism, to see this let xiy j,xlyt Q with i,l are modulo 2n and j,t are modulo 4. Take ∈ n

i j l t i+( 1) jl j+t i+( 1) jl j+t φ (x y x y ) = φ (x − y ) = x − y kn · kn on the other hand

i j l t i j l t i+( 1) jl j+t φ (x y ) φ (x y ) = x y x y = x − y kn · kn ·

28 so φkn is indeed a homomorphism.

Consider the system (Qn,φkn), we will show that this is an inverse system of ﬁnite generalized quaternion groups. First note that φ : Q Q , by our deﬁnition kk k → k this is the identity map. Now, let m < k < n in I, and consider the maps

φ : Q Q mk k → m and

φ : Q Q kn n → k

take φ φ : Q Q , note that mk ◦ kn n → m

(φ φ )(xiy j) = φ (φ (xiy j)) mk ◦ kn mk kn i j = φmk(x y )

= xiy j on the other hand

i j i j φmn(x y ) = x y this shows φ φ = φ , and hence (Q ,φ ) is an inverse system. Deﬁne mk ◦ kn mn n kn

C = ∏Qn n I ∈ and for each n I let π be the projection map ∈ n

π : C Q n → n deﬁne

slimQ = c C φ π (c) = π (c), for all m k n ∈ mk k m ≤ ←−n I ∈

29 and

Q∗ = x¯,y¯ h i wherex ¯ = (xi)i I andy ¯ = (yi)i I, we have ∈ ∈

i ∞ x¯ = LCM(2 )i I = 2 SN and y¯ = LCM(4)i I = 4 | | ∈ ∈ | | ∈

Also

1 1 1 1 1 y¯x¯y¯− = (yi)i I(xi)i I(y− )i I = (yixiy− )i I = (x− )i I = x¯− ∈ ∈ i ∈ i ∈ i ∈

n 2 since x2 − = y2 for all i I, we can write any element in Q asx ¯ay¯b where b is modulo i i ∈ ∗ 4. Now we show that Q∗ = limQn, let n I ←−∈

a b a b φ : Q∗ Q via φ (x¯ y¯ ) = x y n → n n n n and

a b a b θ : Q∗ slimQn via θ(x¯ y¯ ) = (φn(x¯ y¯ ))n I ∈ → n I ←−∈ a b a b a b First we show that θ is injective: Letx ¯ 1 y¯ 1 andx ¯ 2 y¯ 2 be in Q∗, if θ(x¯ 1 y¯ 1 ) =

θ(x¯a2 y¯b2 ) then

a1 b1 a2 b2 (φn(x¯ y¯ ))n I = (φn(x¯ y¯ ))n I ∈ ∈ and so

xa1 yb1 = xa2 yb2 for all n I n n n n ∈

a1 b1 a2 b2 an b this givesx ¯ y¯ = x¯ y¯ . Now we show that θ is subjective, let c = (xn yn)n I, where ∈ n a¯ an is modulo 2 and b is modulo 4. Takea ¯ = ∏ an, and considerx ¯ y¯ Q∗, we have n I ∈ ∈

a¯ b a¯ b an b θ(x¯ y¯ ) = (φn(x¯ y¯ ))n I = (xn yn)n I = c ∈ ∈

this shows that θ is a surjection. Therefore θ is a bijection from Q∗ to limQn, thus n I ←−∈

30 Q∗ = limQn. Thus we have the following n I ←−∈ Example 3.3.1. The proﬁnite generalized quaternion group Q∗ is the inverse limit of ﬁnite generalized quaternion groups.

3.4 Proﬁnite Hamiltonian Groups

Deﬁnition 3.4.1. A ﬁnite group G is an elementary abelian p-group if every non-trivial element has order p.

We can write any elementary abelian group as

(Z/pZ)n

for n Z 0. We note that not every elementary abelian group is cyclic, for example ∈ ≥

Z/3Z Z/3Z × is elementary abelian but not cyclic.

Remark 3.4.2. An elementary abelian pro-2 group is the inverse limit of ﬁnite elementary abelian 2-groups.

Theorem 3.4.3 (Goursat’s Lemma [3]). There is a bijective correspondence between subgroups G of A B and quintuples G¯ ,G ,G¯ ,G , f , where G E G¯ A, G E × { 1 1 2 2 } 1 1 ≤ 2 G¯ B, and f : G¯ /G G¯ /G is an isomorphism. 2 ≤ 1 1 → 2 2

Deﬁnition 3.4.4. A ﬁnite group is said to be Hamiltonian if all of its subgroups are normal.

Let I = N, and let H = Hi i I be a collection of hamiltonian groups, ﬁrst { } ∈ we see that any subgroup of Hi is hamiltonian, since subgroups which are normal in the whole group are also normal in any subgroup that contain them. Let Ni CHi and consider

31 H /N since subgroups are of the form K N /N where K H which is also normal, but i i i i i i ≤ i then K N CH and so K N /N C H /N and so H /N H . Now let K ,L H and take i i i i i i i i i i ∈ i i ∈ K L by Goursat’s lemma we can see that any subgroup of the product is normal and i × i hence K L H . i × i ∈ H = H K H Let ∗ limi I i, any closed subgroup ∗ of ∗ can be viewed as ←− ∈

K∗ lim K∗/(K∗ N) ∩ N←−EoH∗ where K /(K N) is isomorphic to a normal subgroup in H /N for some open normal ∗ ∗ ∩ ∗ subgroup N of H∗. So

h h (K∗) ∗ lim (K∗/(K∗ N)) N lim K∗/(K∗ N) = K∗ ∩ ∩ N←−EoH∗ N←−EoH∗ and so any subgroup of H∗ is normal.

Deﬁne E to be the set of all ﬁnite elementary abelian 2-groups, and Aodd to be the set of all abelian groups in which every element of is of odd prime order. Note that both E and A are closed under subgroups, quotients, and products. For i N, let odd ∈ E E and set i ∈ E∗ = lim Ei i N ←−∈ is an elementary abelian pro-2 group. Also, for A A we have i ∈ odd

A∗ = lim Ai i N ←−∈ is a proﬁnite abelian group in which every element is of odd Steinitz prime order.

Theorem 3.4.5. Let H be a pro-Hamiltonian group. Then H is a profinite abelian group, or H = Q E A where E is an elementary abelian pro-2 group, Q is a profinite ∗ × ∗ × ∗ ∗ ∗ quaternion group and A∗ is a profinite abelian group in which every element is of odd Steinitz prime order.

32 Proof. Any ﬁnite Hamiltonian group is either abelian or can be written as

H = Q E A i i × i × i

where Ei is elementary abelian 2-group, and A is abelian group in which every element is of odd prime order. Taking the inverse limit we have

lim Hi = lim (Qi Ei Ai) i N i N × × ←−∈ ←−∈ since

lim (Ri Si) = lim Ri lim Si i N × i N × i N ←−∈ ←−∈ ←−∈ for Ri and Si are ﬁnite groups, we have

lim Hi = lim Qi lim Ei lim Ai = Q∗ E∗ A∗ i N i N × i N × i N × × ←−∈ ←−∈ ←−∈ ←−∈ so

H∗ Q∗ E∗ A∗ × ×

3.5 Sylow Intersections in Proﬁnite Groups

Definition 3.5.1. Let G be a profinite group, and P and Q be sylow p-subgroups of G such that P , Q. The subgroup P Q of G is called a maximal sylow p-intersection if ∩ whenever P Q R S for sylow p-subgroups R and S of G, we have P Q = R S. ∩ ≤ ∩ ∩ ∩ Lemma 3.5.2. Let G be a profinite group, and let K = (S S Syl (G)). Then: | ∈ p T (a)K E G, and K is a pro-p subgroup of G.

(b) K contains every normal pro-p subgroup of G.

33 Proof. Let g G be arbitrary, since K S for all S Syl (G) we have Kg Sg for all ∈ ≤ ∈ p ≤ S Syl (G), by proposition 2.4.7(d) we have Kg K and so K E G. ∈ p ≤ To show that K is a pro-p subgroup, Let I = (N N / G) be a ﬁlter base such | o that (N N / G) = 1. Then by theorem 2.3.18 we have | o T

K limK/(K N) N I ∩ ←−∈ here clearly K N / K since K E G. and since K/(K N) G/(K N) is a p-subgroup ∩ ∩ ≤ ∩ for all N I, this proves (a). Now, Let H be normal pro-p subgroup of G, since sylow ∈ p-subgroups are maximal pro-p subgroups we have H K. So there exists P syl (G) ≤ ∈ p such that H P. By normality Hg = H for all g G. we have ≤ ∈

H = Hg Pg for all g G ≤ ∈

Thus H Pg = (P P Syl (G)) = K ≤ | ∈ p g G \∈ \ this gives (b).

Theorem 3.5.3. Let G be a proﬁnite group, let D = P Q be a maximal sylow p- ∩ intersection of P,Q Sylp(G) with P , Q. Then: ∈

(a) D is a proper subgroup of each sylow p-subgroup of NG(D).

(b) Any two distinct sylow p-subgroups of NG(D) have intersection D.

taining D and the sylow p-subgroups of NG(D).

Proof. Since D = P Q, where P,Q Syl (G), note that D E N (D). So D is a pro-p ∩ ∈ p G subgroup of NG(D) contained in every sylow p-subgroup of NG(D) . For strict inequality: since D = P Q, we have D < P and this gives D < N (D) N (D), and we have ∩ P ≤ G (a). For part (b) let R1 and S1 be sylow p-subgroups of NG(D) such that R1 , S1, we want

34 to show that D = R S . As D C N (D) we have D R and D S , so D R S . 1 ∩ 1 G ≤ 1 ≤ 1 ≤ 1 ∩ 1 Let R and S be sylow p-subgroups of G such that R R and S S. then D = R S, 1 ≤ 1 ≤ ∩ and D R S R S = D. ≤ 1 ∩ 1 ≤ ∩

Hence D = R S 1 ∩ 1

For part (c), let D = P Syl (G) D P and deﬁne Φ : D Syl (N (D)) by ∈ p ≤ → p G

Φ(P) = P N (D). ∩ G

Let S Syl (N (D)), then there is an S D such that S S and ∈ p G ∗ ∈ ≤ ∗

Φ(S∗) = S∗ N (D) ∩ G but S S N (D) so Φ(S ) = S ≤ ∗ ∩ G ∗

35 Chapter 4

The Transfer Homomorphism for Proﬁnite Groups

4.1 Necessary Background in Set Theory

Let X be a well ordered set, a segment of X is a subset S such that for any a,b X with ∈ a b then a S implies that b S. If S is proper and a is the ﬁrst element of X S then ≤ ∈ ∈ −

S = b X b < a a { ∈ | } called the initial segment. For two well-ordered sets X and Y we say that X is order isomorphic to Y if there is a bijective function f : X Y such that if a b in X then → ≤ f (a) f (b) in Y, we write X Y. An ordinal number α is a well-ordered set if a = S ≤ ≈ a for all a α. If X is a well-ordered set such that X α then we say that X has ordinal ∈ ≈ number α and we write ord(X) = α. If X is finite then α is a finite ordinal otherwise α is an infinite ordinal see [4].

Example 4.1.1 ([4]). Take N = 0,1,2, with the usual order, denote the ordinal { ···} number of N by ω. Let X = a,b,c,d, be a well ordered set with a < b < c < d < { ···} , then S 0 so a occupies the ﬁrst position, likewise b occupies the second position, ··· a ≈ 36 c the third and so on, see [3]. In general:

Theorem 4.1.2 ([2]). Every well-ordered set has a unique ordinal number.

Ordinal numbers can be added, multiplied and exponentiated. The class of ordinal numbers Ord forms a chain

0 < 1 < 2 < < ω < ω + 1 < ω + 2 < < ω2 < ω2 + 1 < ··· ··· ···

For an ordinal α, we call the ordinal number α + 1 a successor of α, and we call α a predecessor of the ordinal number α + 1. An ordinal number is called a limit ordinal number if and only if it does not have a predecessor.

Theorem 4.1.3 (The Principle of Transﬁnite recursion over Ord [4]). Let φ(α) be a statement that is either true or false for each α Ord. If φ(β) is true for each β < α ∈ implies that φ(α) is true, then φ(α) is true for each α Ord i.e. ∈

1. φ(0) is true and if φ(α) is true, then φ(α + 1) is true for each non-limit ordinal α.

2. If β is a limit ordinal and φ(α) is true for each α < β then φ(β) is true.

Note that ordinal numbers capture the position of an element in a set. While similar in construction the cardinal numbers capture the cardinality of a set; i.e the number of elements in this set. The class of cardinal numbers Card form a chain

0 < 1 < 2 < < ℵ < ℵ < ℵ < ··· 0 1 2 ···

where ℵ0 = card(N), ℵ1 = card(P(N)) and ℵ1 = card(P(P(N))) and so on. The following theorems are equivalent.

Theorem 4.1.4 ([2]). Every set has a unique cardinal number.

37 Theorem 4.1.5 (Theorem AC1 [2]). Any collection of nonempty sets has a choice function, i.e Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f deﬁned on C with the property that, for each set S in the collection, f(S) is a member of S.

Theorem 4.1.6 (Theorem CAC [2]). Any collection of mutually disjoint nonempty sets has a transversal.

Theorem 4.1.7 (Zermelo’s Theorem [2]). Every set can be well ordered.

Theorem 4.1.8 ([2]). Every well-ordered set is isomorphic to a unique ordinal number.

4.2 The Permanent Mapping

Deﬁnition 4.2.1. Let A = [a , ] be an (n n)-matrix and S be the symmetric group on i j × n 1,2, ,n . The permanent of A is given as { ··· }

n perm(A) = ∑ ∏ai,σ(i) σ S i=1 ∈ n

Deﬁnition 4.2.2. A monomial matrix A over a commutative ring R with unity, is an (n n) matrix in which each row and each column has exactly one nonzero entry from × − R. It can be written as a product of a permutation matrix P and a diagonal matrix D

A = DP since

Mn(R) = Dn o Σn n × where Dn is the group of all diagonal (n n)-matrices with entries from R, and Σn n is × × the group of all permutation (n n)-matrices. Then ×

n perm(A) = ∏ai where ai is the non-zero entry in row i. i=1

38 Remark 4.2.3. Let A and B be two monomial matrices over a commutative ring with unity, Then perm(AB) = perm(A)perm(B). To see this note that AB is another monomial matrix in which each nonzero entry has the form (ab)i,so

n perm(AB) = ∏(ab)i i=1 by commutativity and associativity

n n perm(AB) = ∏ai ∏bi = perm(A)perm(B) i=1 i=1

39 4.3 The Transfer of Proﬁnite Groups

Let G be a proﬁnite group with H as a closed subgroup. Consider the set

H G = Hg g G \ { | ∈ } of all the right cosets of H in G, we have

G = Hg g G G∈

By theorem 4.2.3, the cardinality of H G is G : H = n SN. Considered as \ | | ∈ an uncountable set we can deﬁne a well-ordering using Zermelo’s theorem. Since it is a collection of mutually disjoint nonempty sets, by theorem 4.2.5 it has a transversal Y which has the same cardinality as H G. Well order Y by the choice function \

ν : H G Y by ν(Hx) = y for x G and y Hx \ → Hx ∈ Hx ∈ it follows that this function ν is an order preserving bijection, using theorem 4.1.7, we can take ∆ SN to index Y ∈ Y = y ∆ { α }α=1 as a ﬁrst element in Y we take idG which corresponds to α = 1. Now, Let (Gi,φi j,I) be an inverse system of ﬁnite groups, set

G = limGi. i I ←−∈

The set of right cosets H G, which is a topological space with respect to the \ quotient topology, is the proﬁnite topological space [25]

H G = limHi Gi \ i I \ ←−∈ 40 The natural group action of G on H G is the continuous map \

φ : G H G H G, with φ(g,Hy ) = (Hy )g where α ∆ × \ → \ α α ∈ this action will give the permutation representation

ρ : G SH G → \ given by

ρ(g)(Hyα ) = (Hyα )g = Hyβ where α,β ∆, note that if g Hyα then α = β otherwise α , β . This permutation ∈ ∈ representation will permute the elements in H G likewise the elements in Y. If g G \ ∈ then

(Hyα )g = Hyβ we have

1 h = y g y− H αβ α β ∈

Deﬁnition 4.3.1. For a Steinitz number n we say that a monomial (n n)-matrix A over × a commutative ring with identity R is known if the limit in transﬁnite induction always exist. i.e

lim ∏ am converges in R i n m i → ≤ and is independent of the order of the indices, where a R is the only nonzero entry in j ∈ the j-th row of A,

Fix an ordering ∆ of Y and let K C H so that H/K is abelian. Deﬁne M∆(H/K) to be the set of all known monomial matrices with entries from H/K and for any A M (H/K) we have exactly one nonzero element in each row and column from ∈ ∆ H/K. For any z H/K we deﬁne 0 z = z 0 = 0 and z + 0 = 0 + z = z. ∈ ∗ ∗

41 Definition 4.3.2. Let G be a profinite group, H be a closed subgroup of G, and K be a closed normal subgroup of H such that H/K is abelian. Let Y = y ∆ be a set of { α }α=1 coset representatives of H in G. The monomial representation of a profinite group G over the abelian group H/K is given by

µ : G M (H/K) via µ(g) = A M (H/K) → ∆ ∈ ∆

where A = [aα,β ] with

1 y g y− K if y g = y α β α · β aαβ =   0 otherwise   Theorem 4.3.3. The monomial representation in the above deﬁnition is a well deﬁned continuous homomorphism.

Proof. To show that it is well deﬁned, let Z = zα α ∆ be another set of coset repre- { } ∈ sentatives of H in G with relation z = u y . For any g G we have α α α ∈

1 1 1 1 zα g zα− K = uα yα g (uα yα )− K = uα (yα g yα− )uα− K then

1 1 1 1 zα g zα− K = uα K yα g yα− K uα− K = yα g yα− K. To show that it is a homomorphism, let g1,g2 be in G, and call µ(g1) = A1 and µ(g ) = A . Then µ(g )µ(g ) = A A M (H/K) is another monomial matrix 2 2 1 2 1 2 ∈ ∆ 1 in which each nonzero entry is given as yα g1g2 yγ− K, to see this take A1 = [aα,β ] and

A2 = [bδ,γ ] where

1 y g y− K if y g = y α 1 β α · β aαβ =   0 otherwise   42 and y g y 1K if y g = y δ 2 γ− δ · γ bδγ =   0 otherwise  for g and g in G and α,β,δ,γ ∆. Since A and A have exactly one nonzero entry in 1 2 ∈ 1 2 each row and each column we have for A1A2 = [aα,β bδ,γ ] where β = δ, so the nonzero entries of A1A2 look like

1 1 1 1 1 yα g1 yβ− K yβ g2 yγ− K = yα g1 yβ− yβ g2 yγ− K = yα g1g2 yγ− K then µ(g1g2) = µ(g1)µ(g2). Finally, multiplication in H/K is continuous since H/K is a proﬁnite group.

Deﬁnition 4.3.4. Let G be a proﬁnite group and H be a closed subgroup of G with index ∆ in SN. Let K be a normal subgroup of H such that H/K is abelian. The permanent of a monomial matrix A M (H/K) the known monomial matrices is the map ∈ ∆

perm : M (H/K) H/K ∆ ∆ → where

M∆(H/K) = D∆ o Σ∆ ∆ × and D∆ is the set of all ∆ ∆ diagonal matrices, and Σ∆ ∆ is the set of all ∆ ∆ × × × permutation matrices. Which satisﬁes

perm∆(A) = ∏ aα H/K (4.1) α ∆ ∈ ∈ and

perm∆(A) = ∏ aαβ H/K (4.2) (α,β) ∆ ∆ ∈ ∈ × aαβ ,0 where a = a H/K, and the products are commutative and associative. αβ α ∈

43 Theorem 4.3.5. Let G be a proﬁnite group, H a closed subgroup and K be a closed normal in H such that H/K is abelian. Then perm∆ is a a continuous homomorphism into H/K.

Proof. To show that perm∆ is a homomorphism, let A and B be in M∆(H/K), using equation (4.1) we have

perm∆(A) = ∏ aα = ∏ aαβ α ∆ (α,β) ∆ ∆ ∈ ∈ × aα,β ,0 and

perm∆(B) = ∏ bδ = ∏ bδγ δ ∆ (δ,γ) ∆ ∆ ∈ ∈ × bδγ ,0 then

perm∆(A)perm∆(B) = ∏ aα ∏ bδ = ∏ aαβ ∏ bδγ α ∆ δ ∆ (α,β) ∆ ∆ (δ,γ) ∆ ∆ ∈ ∈ ∈ × ∈ × aαβ ,0 bδγ ,0 hence

perm∆(A)perm∆(B) = ∏ aαβ bβγ (α,β),(β,γ) ∆ ∆ ∈ × aαβ ,bβγ ,0 on the other hand for AB we have

perm∆(AB) = ∏ aαβ bβγ (α,β),(β,γ) ∆ ∆ ∈ × aαβ ,bβγ ,0 hence

perm∆(AB) = perm∆(A)perm∆(B)

Finally, since multiplication is continuous, then perm∆ is continuous.

Deﬁnition 4.3.6. Let G be a proﬁnite group and H be a closed subgroup of G such that G : H = n, with n SN, let H = [H,H]. Let Y be a transversal of H in G with | | ∈ 0

44 Y = ∆ SN. The transfer map from G into H/H is the composition | | ∈ 0

G 1 VH (g) = perm∆(µ(g)) = ∏ yα g yβ− H0 α,β ∆ ∈

Theorem 4.3.7. The transfer map from G into H/H0 is a well deﬁned continuous homomorphism.

Proof. We have seen that perm∆ is a continuous homomorphism and µ is a well deﬁned continuous homomorphism, the result follows.

Let G be a proﬁnite group, H is a closed subgroup with G : H SN and let | | ∈

Y = yα α ∆ to be a right transversal of H in G with Y = ∆, let H0 be the characteristic { } ∈ | | group of H. For g G let αg to be in ∆ such that y g y 1 H, we can write ∈ α α−g ∈

G 1 1 VH (g) = perm∆(µ(g)) = ∏ yα g yα−g H0 = ∏ yα g yα−g H0 yα ,yαg Y α ∆ ∈ ∈

If we ﬁx g we get the permutation π(g) : α αg, suppose that π(g) decomposes → into Γ disjoint cycles C (g) where γ Γ ∆, each cycle consists of members in ∆, γ ∈ ≤ deﬁne

Cset(g) = ε ∆ ε appears in C (g) γ ∈ γ these sets partition ∆ i.e

set ∆ = Cγ (g). γ Γ G∈ Each cycle has a length r SN which also can be thought as an ordinal number, γ ∈ and for γ Γ take y gκ 0 κ < r to be the g orbits (or cycles) where y grγ Hy ∈ γ ≤ γ γ ∈ γ κ and yγ g < Hyγ for any κ < rγ . Then

κ G = Hyγ g γ Γ κ

45 Lemma 4.3.8. From the construction above we have

G rγ 1 VH (g) = ∏ yγ g yγ− H0 γ Γ ∈

rγ 1 1 where yγ g yγ− is the ﬁrst power of yγ g yγ− that lies in H.

Proof. For g G we have a disjoint coset decomposition ∈

κ G = Hyγ g γ Γ κ

r 1 r y g γ − g = y g γ Hy γ γ ∈ γ and for any κ < r 1 we have γ −

y gκ g = y gκ+1 Hy gκ+1 γ γ ∈ γ

G Consider the transfer VH (g) we have

G 1 VH (g) = ∏ yα g yβ− H0 y ,y Y α β ∈ where yβ = ν(yα g). In calculating the transfer we have

κ κ+1 1 κ+1 κ+1 1 1 yγ g g yγ g − = yγ g g − yγ− = 1 for any κ < r 1, and γ − rγ 1 rγ 1 rγ 1 yγ g − g yγ g − = yγ g yγ− then

G rγ 1 VH (g) = ∏ yγ g yγ− H0 γ Γ ∈ 46 Theorem 4.3.9 (Transitivity of the Transfer). Let G be a proﬁnite group, and let H and K be closed subgroups such that K H G with each index is in SN. Then for all ≤ ≤ g G we have ∈ G H G VK (g) = VK VH (g) Proof. Let Y be a right transversal of H in G of cardinality ∆ and Z be a right transversal of K in H of cardinality Γ , then

G = Hyα and H = Kzδ α ∆ δ Γ G∈ G∈ then

G = Kzδ yα . α ∆ δ Γ G∈ G∈ To ﬁnish off the proof we need to show that for any z Z and y Y that ∈ ∈ (z y ) g (z y ) 1 K. This is the same as showing that ZY is a right transversal γ α δ β − ∈ of K in G; where ZY = zy z Z, y Y , let g G then Hg = Hy for some y Y, { ∈ ∈ } ∈ ∈ so gy 1 H. Then K(gy 1 ) = Kz for some z Z, we have Kg = K(zy) this shows − ∈ − ∈ that zy Kg. We claim that this element zy is the unique element of ZY in Kg; since ∈ otherwise if Kz y = Kz y , where z ,z Z and y ,y Y, but then Ky = Ky γ α δ β γ δ ∈ α β ∈ α β since Kz ,Kz H, so we have y = y since Y is a right transversal of H is G. Thus γ δ ⊂ α β Kzγ yα = Kzδ yα therefore Kzγ = Kzδ , but Z is a right transversal of K in H we have zγ = zδ . Hence ZY is a right transversal of K in G. Now we show that

1 (z y ) g (z y )− K. γ α δ β ∈

From the above claim, we have K(z y )g = Kz y , and since Kz ,Kz H we γ α δ β γ δ ⊂ 1 have Hy g = Hy , so y g = y and there is some h H such that h = y g y− So α β α · β ∈ α β

1 1 Kzγ k = Kzγ yα g yβ− = Kzδ yβ yβ− = Kzδ

47 1 so z k = z and there is some kinK such that k = z kz− . Finally γ · δ γ δ

1 1 1 1 (zγ yβ )g(zδ yβ )− = zγ yβ gyβ− zδ− = zγ kzδ− = h so for any g G we have y g = h y , but h H then z g = k z , then ∈ α αβ β αβ ∈ δ αβδγ γ

zδ yα = kαβδγ zγ yβ where k K. Hence for g G we have αβδγ ∈ ∈

G VK (g) = ∏ kαβδγ K0 α ∆,δ Γ ∈ ∈ and

G VH (g) = ∏ hαβ H0 α ∆ ∈ but

H VK (hαβ ) = ∏ kαβδγ K0 δ Γ ∈ then

G H H H G VK (g) = ∏ VK (hαβ ) K0 = VK ∏ hαβ K0 = VK VH (g) α ∆ α ∆ ! ∈ ∈

4.4 Applications of the Proﬁnite Transfer

The ﬁrst application is Burnside’s theorem for proﬁnite groups, which is an analog of Burnside’s theorem in [14]. we will prove it using the transfer homomorphism.

Theorem 4.4.1 (Burnside’s Normal p-Complement Theorem for Proﬁnite Groups [14]). Let G be a proﬁnite group and P is a sylow p-subgroup of G be such that P Z(NG(P)). ≤ Then G has a closed normal subgroup H which has the elements of P as its coset representatives.

48 We note that this theorem was proved for proﬁnite groups in the context of fusion in [10]. To prove the theorem we need the following lemma, which is proved for ﬁnite groups in [14].

Lemma 4.4.2. Let G be a proﬁnite group, if two subsets K1 and K2 of G are normal in a sylow pro-p subgroup P of G and are conjugate in G then K1 and K2 are conjugate in

NG(P).

Proof. Suppose that Kx = K for some x G. Since K E P then K = Kx is normal 1 2 ∈ 1 2 1 x in P = Q. So both P and Q are contained in the normalizer of K2. As sylow pro-p subgroups are conjugate in N (K ). So Qy = P for some y G with Ky = K . Now , let G 2 ∈ 2 2 z z z = xy we have P = P, K1 = K2. This ends the proof of the lemma.

here is a proof given by an axiomitized transfer.

Proof. First note that P is abelian so P0 = 1, as a closed subgroup we have

G = Pyα α ∆ G∈ where Y is a transversal of P in G with cardinality ∆. Let u P and consider the orbits ∈ κ r κ of u in G given by yγ u 0 κ < rγ where γ Γ ∆ and yγ u γ Pyγ and yγ u < Pyγ { ≤ } ∈ ≤ ∈ for any κ < r 1. Consider the transfer homomorphism from G to P, we have γ −

G 1 1 VP (u) = ∏ yα u yβ− P0 = ∏ yα u yβ− α,β ∆ α,β ∆ ∈ ∈ as P0 = 1. View the permutation u on P in terms of the cycle structure, in calculating the transfer we have

κ κ+1 1 κ+1 κ+1 1 1 yγ u u yγ u − = yγ u u − yγ− = 1 for any κ < r 1, and γ − rγ 1 rγ 1 rγ 1 yγ u − u yγ u − = yγ u yγ− 49 then

G rγ 1 VP (u) = ∏ yγ u yγ− γ Γ ∈ Now, y urγ y 1 P is conjugate to urγ in G, let us consider K = y urγ y 1 γ γ− ∈ 1 { γ γ− } and K = urγ , since P is abelian K and K are normal in P, using the last lemma 2 { } 1 2 we have y urγ y 1 = z urγ z 1 with z N (P). By hypothesis P in in the center of its γ γ− − ∈ G r 1 r normalizer, this means that zu γ z− = u γ . Hence

G rγ VP (u) = ∏ u γ Γ ∈ and since r where γ Γ are the lengths of the cycles that partitioned ∆ we have γ ∈

G ∆ G:P VP (u) = u = u| |

Since P is a pro-p sylow subgroup of order pm(p), with m(p) ∞, we have ≤ p - G : P . Thus in the transfer of G onto P, V G(P) = P because trivially the transfer | | P G cannot be larger than P we have VP (G) = P. The kernel of this homomorphism must be a subgroup, H, of index pm(p) in G and of order G : P . Hence H is a normal subgroup | | of index pm(p) in G and so the elements of P can be used as a transversal of H in G.

The next result is the focal subgroup theorem for proﬁnite groups, again we will use the transfer in the proof.

Theorem 4.4.3. Let G be a proﬁnite group, P a pro-p sylow subgroup of G and G0 = [G,G]. Then

G V (G) P/P G0 P ∩

Proof. Let Y = y ∆ where ∆ = G : P be a transversal of P in G. Consider the { α }α=1 | | transfer homomorphism V G : G P/P and note that since P/P is a p-group then for P → 0 0 any x G with order prime to p we have V G(x) = 1. Since G is generated by P and the ∈ P other sylow subgroups for other primes we have V G(G) = V G(P). Let u P by lemma P P ∈

50 4.3.9 we have

G rγ 1 VP (u) = ∏ yγ u yγ− P0 γ Γ ∈ with γ Γ ∆. Note that ∈ ≤ rγ 1 rγ rγ 1 yγ u yγ− = u [u ,yγ− ]

rγ 1 since [u ,yγ− ] is in G0 we have

G rγ G:P VP (u) = ∏ u G0 = u| | G0. γ Γ ∈

G G Since p, G : P = 1 then V (u) . 1 for u P and u < G0. We also have V (G0) 1 | | P ∈ P ≡ since VG(G) is abelian. Hence the kernel of P V G(P) is P G , then P → P ∩ 0

G V (G) P/P G0 P ∩

We present the following as a beginning effort to prove M. Hall Theorem in [14].

Deﬁnition 4.4.4. The diagonal contribution, d(g) is the contribution of cycles of length G one in π(g) to VH (g), and is given by

1 d(g) = ∏ yα g yα− H0 α=αg

Remark 4.4.5. Note that d(g) = ∏α ∆ hαα H0 where π(g) = hαβ H0 (α,β) ∆ ∆. ∈ ∈ × Lemma 4.4.6. Let G be a proﬁnite group, if u and v are conjugate in G, then d(u) =

1 1 d(v), and d(u− ) = [d(u)]− .

Proof. Let v = t 1ut for some t G. From α = αu we have α = αtvt 1 which gives − ∈ − αt = αtv, this means that α = αu is equivalent to αt = αtv. By deﬁnition

1 1 d(v) = ∏ yαtvyα−t H0 = ∏ yαtvyα−t H0 αt=αtv α=αu

51 1 1 1 1 since v = t− ut we have v = t− yα− yα uxα− xαt we have

1 1 1 1 d(v) = ∏ yαtt− yα− yα uyα− yαtyα−t H0 α=αu 1 1 1 but yαtt− yα− and yαtyα−t are in H we obtain

1 d(v) = ∏ yα uyα− H0 = d(u) α=αu

1 since α = αu is equivalent to α = αu− we have

1 1 1 1 1 1 d(u− ) = ∏ yα u− yα− H0 = ∏ yα uyα− − H0 = [d(u)]− α=αu α=αu

Definition 4.4.7. Let G be a profinite group and H be a closed subgroup of G. For h H ∈ define

1 d∗(h) = h− d(h)

Lemma 4.4.8. Let G be a proﬁnite group and H be a closed subgroup of G. If h H ∈ then

G G:H rγ rγ 1 VH (h) = h| | ∏ d∗(h ) d∗(yγ h yγ ) − γ Γ ∈ 1 1 Proof. From d∗(h) = h− d(h) we have h = d(h) d∗(h) − and by lemma 4.4.5 h = 1 d(h)d∗(h− ). Again by lemma 4.4.5 we have

rγ rγ d(h ) = d(yγ h yγ )

r hence if yγ h γ yγ is in H we have

rγ rγ rγ 1 rγ rγ rγ 1 yγ h yγ = d(h ) d∗(yγ h yγ ) − = h d∗(h ) d∗(yγ h yγ ) −

52 by lemma 4.3.9 we have for h H ∈

G rγ 1 VH (h) = ∏ yγ g yγ− H0 γ Γ ∈ but

rγ rγ rγ 1 rγ rγ rγ 1 yγ h yγ = d(h ) d∗(yγ h yγ ) − = h d∗(h ) d∗(yγ h yγ ) − . then

G rγ rγ rγ 1 G:H rγ rγ 1 VH (h) = ∏ h d∗(h ) d∗(yγ h yγ ) − H0 = h| | ∏ d∗(h ) d∗(yγ h yγ ) − γ Γ γ Γ ∈ ∈

Corollary 4.4.9. If d (h) is in H for any h H then V G(h) = h G:H . ∗ 0 ∈ H | |

Proof. This is an immediate consequence of the last lemma.

Remark 4.4.10. All results on the diagonal contribution is work towards the proof of Philip Hall Theorem in [14]

53 Chapter 5

Future Work

First, can we deﬁne the transfer without axiomatizing? That is: Is the set of known (n n)-monomial matrices the set of (n n)-monomial matrices? × ×

In addition to many well known results in [14], [27], [11] and [24] as an application to the transfer for finite groups, we will list some of these results for future work. In chapter 14 of [14], Marshall Hall used the transfer as a tool to provide and prove some important results, some of which were extended to profinite groups in chapter 4. Here we list other results of chapter 14 in [14] which we like to extend to profinite groups, these results were listed in chapter 2. We would like to provide an analog of the results by Yoshida [26] in section 4 we wish to extend to profinite groups. let G be a profinite group, and let H,K be closed subgroups of G, by Rep(H/G K) we mean a complete set of double coset representatives \ in G.

Deﬁnition 5.0.1. Let G be a proﬁnite group, S a closed subgroup of S, and W closed normal subgroup of S.

• W is a weakly closed subgroup of S with respect to G if for any g G with W g S ∈ ≤ we have W g = W.

54 • W is strongly closed subgroup of S with respect to G if for any x G we have ∈ W g S W. ∩ ≤

Deﬁnition 5.0.2. Let G be a proﬁnite group, and H be closed subgroups of G, and S be closed subgroups of H. We say that S is of sylow-type in H with respect to G if any G-conjugate of S contained in H is also an H-conjugate to S.

Example 5.0.3. If S is weakly closed subgroup of a sylow subgroup of H, then S is of sylow type in H with respect to G.

Theorem 5.0.4. Let G be a proﬁnite group, and P a sylow p-subgroup of G that is weakly regular. Then P G = P N (P) . ∩ 0 ∩ G 0

Theorem 5.0.5. Let G be a proﬁnite group, and P a sylow p-subgroup of G. Let Q Cc P H G such that Q is of sylow type in H with respect to G and that [x,y; p 1] ≤c ≤c − ∈ Φ (Q) for each x P and y Q. Then P G = P H P N (Q) . ∗ ∈ ∈ ∩ 0 ∩ 0 ∩ G 0

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