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Groups with Conditions on Non-Permutable Subgroups

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Groups with Conditions on Non-Permutable Subgroups GROUPS WITH CONDITIONS ON NON-PERMUTABLE SUBGROUPS by LAXMI KANT CHATAUT MARTYN R. DIXON, COMMITTEE CHAIR MARTIN J. EVANS JON M. CORSON DREW LEWIS RICHARD BORIE A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2015 Copyright Laxmi Kant Chataut 2015 ALL RIGHTS RESERVED ABSTRACT In this dissertation, we study the structure of groups satisfying the weak minimal condi- tion and weak maximal condition on non-permutable subgroups. In Chapter 1, we discuss some definitions and well-known results that we will be using during the dissertation. In Chapter 2, we establish some preliminary results which will be useful during the proof of the main results. In Chapter 3, we express our main results, one of which states that a locally finite group satisfying the weak minimal condition on non-permutable subgroups is either Chernikov or quasihamiltonian. We also prove that, a generalized radical group sat- isfying the weak minimal condition on non-permutable subgroups is either Chernikov or is soluble-by-finite of finite rank. In the Final Chapter, we will discuss the class of groups satisfying the weak maximal condition on non-permutable subgroups. ii DEDICATION I would like to dedicate this dissertation to my parents. First, to my deceased mother who taught me invaluable lessons in life. Second to my father, who always provided me unconditional moral as well as financial support. iii LIST OF ABBREVIATIONS AND SYMBOLS hxi Subgroup generated by x Aut G The automorphism group of G CG(H) Centralizer of the subgroup H in G NG(H) Normalizer of H in G jG : Hj Index of H in G G0 = [G; G] Derived subgroup of group G H ¡ G H is a normal subgroup of the group G T (G) The torsion subgroup of the group G Gp The p-primary component of the group G Dr Aλ The direct product of the groups fAλjλ 2 Λg λ2Λ Q Hλ The product of the groups fHλjλ 2 Λg λ2Λ rp(G) The p-rank of the group G r0(G) The torsion-free rank of the group G r(G) The rank of the group G CoreH (G) Core of H in G H per GH is permutable in G o Semi direct product ζ(G) Center of the group G Cp1 Quasicyclic p-group, or Pr¨ufer p-group iv ACKNOWLEDGMENTS My deepest gratitude is to my advisor Dr. Martyn R. Dixon for his valuable guidance and advice throughout the course of dissertation. I was fortunate enough to have an advisor like him. His patience, motivation and immense knowledge helped me all the time in my research work and writing of this dissertation. If it wasn't for him this dissertation would not be possible. Besides my advisor, I would like to take this opportunity to thank and appreciate the rest of my dissertation committee: Dr. Martin J. Evans, Dr. Jon Corson, Dr. Drew Lewis and Dr. Richard Borie for their insightful comments and encouragement. I would like to thank the University of Alabama for supporting me financially during the studies. My appreciation also goes to the Chair of the Department of Mathematics, Dr. Wei Shen Hsia, and the Graduate Advisor of the Department of Mathematics, Dr. Vo Liem for their help and contribution, starting from my application process for the Ph.D. program, until the preparation of this dissertation. I must acknowledge as well the many friends, colleagues, students, and professors, who assisted, advised, and supported me in one way or other during my studies back in Nepal as well as at the University of Alabama over the years. I would also like to thank my family to whom I owe a great deal. To my late mother, Janaki Chataut, thank you for teaching me important lessons in life. To my father, Kr- ishna Datta Chataut, for always prioritizing my education and providing me the best of the opportunities even during the time of financial difficulties. My elder brother Ramesh Chataut deserves appreciation for his continuous financial and moral support during my stay in Kathmandu for the higher education. Also thanks to my sister Hemlata, other siblings and sister-in-laws for always being by my side whenever I needed them. v Last but not least, none of this would have been possible without the love and patience of my better half Sara Chataut. Her generosity, understanding and motivation during the course of writing this dissertation was immense. vi CONTENTS ABSTRACT ..........................................................................ii DEDICATION . iii LIST OF ABBREVIATIONS AND SYMBOLS . iv ACKNOWLEDGMENTS . v 1. INTRODUCTION . .1 1.1.Permutable Subgroups . 2 1.2. Min-P and Max-P ...............................................................5 1.3. Weak-Min-P and Weak-Max-P ................................................13 1.4. Groups of Finite Rank . .17 2. PRELIMINARY RESULTS . .20 3. PROOF OF MAIN THEOREMS . 32 4. GROUPS WITH THE WEAK MAXIMAL CONDITION ON NON-PERMUTABLE SUBGROUPS . 46 REFERENCES . 55 vii CHAPTER 1 INTRODUCTION The structure of infinite groups satisfying the minimal (respectively maximal) condition or the weak minimal (respectively weak maximal) condition has been one of the important aspects for the development of infinite group theory. If P is a subgroup theoretical property or class of groups then P¯ denotes the class of all groups that are either not P-groups or are trivial. In 1964, S. N. Chernikov started the investigation of groups satisfying the minimal condition on subgroups which do not have the property P, denoted by min-P¯, in a series of articles (for a general reference, see [4]). In particular, he studied groups satisfying the minimal condition on non-abelian subgroups and groups satisfying the minimal condition on non-normal subgroups. He proved that if such a group has a series with finite factors, then either the group satisfies the minimal condition on all subgroups or all the subgroups have the prescribed property P. Later, Philips and Wilson [19] investigated the same type of problem for several different choices of the property P, and proved in particular that a locally graded group satisfying the minimal condition on non-normal subgroups either is a Chernikov group or has all its subgroups normal. Taking motivation from such work, in their paper [3], M.R. Celentani and Antonella Leone considered the class of groups satisfying the minimal condition on non-quasinormal subgroups. The structure of such groups was given by the following result: Theorem 1.1. [3, Theorem C] Let G be a group which either is non-periodic or locally graded. If G satisfies the minimal condition on non-quasinormal subgroups, then either G is quasihamiltonian or it is a Chernikov group. The concept of the weak minimal condition was introduced by D. I. Zaitsev [24]. In [25], he investigated the class of groups satisfying the weak minimal condition on non-abelian 1 subgroups where he was able to show that in the case of locally soluble-by-finite (also known as locally almost soluble) groups the weak minimal condition on non-abelian subgroups is equivalent to the weak minimal condition on subgroups. By virtue of the result in [25] such a group is then a soluble-by-finite minimax group. Groups with the weak minimal condition on non-normal subgroups and non-subnormal subgroups were the subject of interest in [13] and [14] respectively. In [14], the authors were able to establish the following theorem: Theorem 1.2. [14, Theorem C] Let G be a generalized radical group with weak minimal condition on non-subnormal subgroups. Then either G is soluble-by-finite and minimax or every subgroup of G is subnormal. The results in these papers and similar other papers were enough to motivate us to investigate the structure of groups satisfying the weak minimal condition on non-permutable subgroups. The main purpose of this dissertation is to study the structure of generalized radical groups satisfying the weak minimal condition on non-permutable subgroups, which we will discuss in Chapter 3. In the coming sections of this chapter, we will go through some important definitions, and results which will be useful in the further chapters and which will also explain some of the terminology used in the Introduction. 1.1 Permutable Subgroups In this first section we discuss the notion of permutable subgroups. We begin the section with the definition of permutable subgroup. Definition 1.1. A subgroup H of a group G is said to be permutable or quasinormal in G if HK = KH for every subgroup K of G. This is equivalent to affirming that HK is a subgroup of G. We often write H per G for H is permutable in G. The concept of permutability was introduced by Ore [17], who called permutable sub- groups quasinormal. Later S. E. Stonehewer introduced the term permutable and the be- havior of permutable subgroups was later investigated by various authors. Obviously, every 2 normal subgroup is permutable but the converse is not true. This can be seen by let- ting the group G be the semidirect product of a cyclic group hxi of order p2 and a cyclic group hyi of order p acting non-trivially. In other words: G is a group of order p3 gen- erated by x and y satisfying the relations xp2 = yp = 1 and y−1xy = xp+1. That is, G = hx; y j xp2 = yp = 1; y−1xy = xp+1i. We claim that the subgroup hyi is permutable in G, but it is not normal. Let H be the subgroup generated by the elements of order p.
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