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The Automorphism Group of the Wreath Product of Finite Groups Western Michigan University ScholarWorks at WMU Dissertations Graduate College 8-1974 The Automorphism Group of the Wreath Product of Finite Groups Kenneth G. Hummel Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Mathematics Commons Recommended Citation Hummel, Kenneth G., "The Automorphism Group of the Wreath Product of Finite Groups" (1974). Dissertations. 2917. https://scholarworks.wmich.edu/dissertations/2917 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]. THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT OF FINITE GROUPS by Kenneth G. Hummel A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the Degree of Doctor of Philosophy Western Michigan University Kalamazoo, Michigan August 1974 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS Since it is impossible to list everyone who has helped, one way or another, along the road to completion of this dissertation, I choose to mention here, since the choice is mine, only the following people. I thank Joe Buckley, under whose able guidance and direction this thesis was developed; my wife Darla and my children, Lacinda and Melinda, who bore graciously both the good moods and bad which I alternately thrust upon them during the period of my work at Western Michigan Uni­ versity; and lastly, the mathematics faculty and depart­ ment for their support, intellectual, moral, and financial. Kenneth G. Hummel Reproduced with permission of the copyright owner. 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Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. | 74-26,836 | HUMMEL, Kenneth 6., 1943- f; THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT | OF FINITE GROUPS. | Western Michigan University, Ph.D., 1974 Mathematics University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS CHAPTER PAGE I PRELIMINARIES ....................... 1 Introduction ...................... 1 Definitions and Notation .......... 3 II THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT .......................... 6 Introduction ...................... 6 Aut(A Wr B) as a Split Extension . 7 The Subgroup H of Aut(A Wr B) . 16 III THE AUTOMORPHISM GROUP OF THEWREATH PRODUCT OF TWO P - G R O U P S ............ 41 Introduction ...................... 41 The Subgroup H .....................42 Some Results on the Structure of Aut (A wr B) .......................53 IV THE CONJECTURE " |g | DIVIDES | A U T G | " .............58 Introduction ...................... 58 G as a Central P r o d u c t .............58 G as a Wreath Product.............. 64 i i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT OF FINITE GROUPS Kenneth G. Hummel, Ph.D. Western Michigan University, 1974 An important concept in group theory is that of the automorphism group of a group. It has been shown that the automorphism group of the standard wreath product W of a group A by a group B is, except in a few cases, a splitting extension of A*HI^ by B*, where A* and B* are isomorphic to the automorphism groups of A and B respectively, 1^ is a particular subgroup of the group of inner automorphisms of W, and H, although somewhat described, is really the "unknown ingredient" of Aut W. In this thesis the subgroup H of Aut W is des­ cribed when the group A is purely non-abelian, abelian, or cyclic, and when both A and B are p-groups. Come of these descriptions yield formulas for the order of H, o(H), (hence for o(Aut W)) and methods whereby all the automorphisms of W may be constructed. Furthermore, when both A and B are p-groups, a Sylow p-subgroup of Aut W is described as a splitting extension. A similar result is obtained for Out W. The conjecture that o(G) divides o(Aut G) when G is a non-cyclic p-group of order greater than p 2 is investigated when G is a central product and also when Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G is a wreath product. Both investigations yield new classes of groups for which the conjecture is true. Fur­ thermore, the central product work shows that the general conjecture will be established if it can be proven for a smaller class of groups. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I PRELIMINARIES Introduction Most branches of mathematics introduce new and impor­ tant concepts into their fields of study when they define "products of" and "operations on" particular objects in their realm. In analysis, for example, products of func­ tions, spaces, and measures are important concepts as well as operations such as differentiation, integration, and summation. In graph theory there is the cartesian product of graphs and operations such as complement and converse. This thesis is concerned primarily with the standard wreath product of groups (which is again a group) and the operation "Aut" which assigns to each group G another group, denoted by Aut G, consisting of all the automorphisms of G. Chapters II and III investigate the automorphism group of the standard wreath product, the combination of these two important group theoretic ideas. A description of this automorphism group has been given by Houghton [9], but his description contains a subgroup H about which little was known. Let W be the standard wreath product of a group A by a group B. Chapter II contains a more detailed description of the subgroup H of Aut W when 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A is purely non-abelain, abelian, or cyclic. Chapter III is an investigation of Aut W when both A and B are finite p-groups. In this case a formula for the order of H, o(H), and hence for o(Aut W) is determined. Fur­ thermore, the proof is constructive and yields a method whereby all the automorphisms in H and hence in Aut W may be realized from certain homomorphisms. Chapter IV investigates the conjecture that o(G) divides o(Aut G) when G is a non-cyclic p-group of or- 2 der greater than p . The conjecture is shown to be true whenever G is the central product of non-trivial proper subgroups A and K, where A is abelian and o(K) di­ vides o(Aut K) . This result extends the classes of groups for which the conjecture is known to be true and also shows that if the conjecture can be proven for a par­ ticular class of p-groups, then it is true for all p-groups under consideration. Also in this chapter, results from Chapters II and III are used to yield theorems giving con­ ditions on the p-groups A and B under which the con­ jecture is true for W = A wr B . Examples of these con­ ditions are: o (A) divides o(Aut A) and o (B) ;> o(Z(A)), where Z(A) is the center of A . Furthermore, it is shown that the conjecture is true for the standard wreath product of any two non-trivial p-groups A and B if a weaker statement is true for all non-abelian indecompos­ able p-groups G . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Definitions and Notation If A and B are non-trivial groups, the wreath product of A by B is defined as follows.
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