Western Michigan University ScholarWorks at WMU Dissertations Graduate College 12-2020 On the Local Theory of Profinite Groups Mohammad Shatnawi Western Michigan University, [email protected] Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Algebra Commons, and the Number Theory Commons Recommended Citation Shatnawi, Mohammad, "On the Local Theory of Profinite Groups" (2020). Dissertations. 3691. https://scholarworks.wmich.edu/dissertations/3691 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 by 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. (c) There exists a one-to-one correspondence between sylow p-subgroups of G con- 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 stan- dard 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.