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CCoolllleeggee ooff SSaaiinntt BBeenneeddiicctt aanndd SSaaiinntt JJoohhnn''ss UUnniivveerrssiittyy DDiiggiittaallCCoommmmoonnss@@CCSSBB//SSJJUU Mathematics Student Work Mathematics 2012 NNoonn--AAbbeelliiaann GGrroouuppss wwiitthh PPeerrffeecctt OOrrddeerr SSuubbsseettss Hongying Zhao Follow this and additional works at: https://digitalcommons.csbsju.edu/math_students Part of the Mathematics Commons RReeccoommmmeennddeedd CCiittaattiioonn Zhao, Hongying, "Non-Abelian Groups with Perfect Order Subsets" (2012). Mathematics Student Work. 2. https://digitalcommons.csbsju.edu/math_students/2 This Thesis is brought to you for free and open access by DigitalCommons@CSB/SJU. It has been accepted for inclusion in Mathematics Student Work by an authorized administrator of DigitalCommons@CSB/SJU. For more information, please contact [email protected]. NON-ABELIAN GROUPS WITH PERFECT ORDER SUBSETS Hongying Zhao Thesis Submitted to The College of Saint Benedict Department of Mathematics for the Degree of Bachelor of Art in Mathematics Thesis Advisor: Dr. Bret Benesh 2012 ABSTRACT Thepurposeofthispaperistoexplorenon-abelianﬁnitegroupswithper- fect order subsets. A ﬁnite groups is said to have perfect order subsets (POS) if thenumberofelementsofeachgivenordercandividetheorderofthegroup. The study of such groups was initiated by Carrie E. Finch and Lenny Jones. In this paper, we construct POS-groups by considering semi-direct products ofcyclicgroups(andsometimesquaternions). i ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Dr. Bret Benesh for his unfailing support and guidance. He has taught me, both consciously and subconsciouly, how to think mathematically. His enthusiasim for math is motivationalandcontagious. HehasbeenmyinspirationasIhurdleallkinds ofobstaclesinthecompletionofmyresearchwork. Iamtrulythankfulforall hiscontributionoftimeandideas. I would also like to thank Dr. Tom Sibley, Dr. Michael Gass and Dr. JenniferGalovichfortheirsteadfastencouragmentandinsightfulsuggestions. Finally,noneofthiswouldhavebeenpossiblewithouttheCollgeofSaint Benedict / Saint John’s University Mathematics Department and Undergrau- date Research program. I sincerely appreciate the opportunity and funding theyprovidedtomakemyundergraduateexperiencestimulating. ii Contents 1 INTRODUCTION 1 1.1 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 NOTATIONSANDNEWLYINTRODUCEDDEFINITIONS . . 2 1.3 BASICEXAMPLESANDNON-EXAMPLES . . . . . . . . . . . 3 2 ABELIANPOSGROUPS 5 3 EULERPHIFUNCTION 7 4 ANINTRODUCTIONTOSEMIDIRECTPRODUCTS 9 5 NONABELIANGROUPSHAVINGPERFECTORDERSUBSETS 12 5.1 SOMEPROPERTIESOFSEMIDIRECTPRODUCTSWITHPER- FECTORDERSUBSETS . . . . . . . . . . . . . . . . . . . . . . . 12 5.2 SEMIDIRECTPRODUCTSWITHINVERSIONS . . . . . . . . . 14 6 OPENQUESTIONS 26 A SEMIDIRECTPRODUCTSWITHNON-INVERSIONS 27 B SEMIDIRECTPRODUCTSWITHTHEQUATERNIONGROUP 31 iii C GAPCODE 37 iv CHAPTER 1 INTRODUCTION Ingrouptheory,therearealotofconnectionsamongordersofgroupsand orders of their subgroups. Among all of them, the most commonly used and well-known one is Lagrange’s Theorem, which states a relationship between theorderofaﬁnitegroupandtheorderofeverysubgroup. InthepaperACu- riousConnectionBetweenFermatNumbersandFiniteGroups,CarrieE.Finchand LennyJonesstudiedgroupswiththepropertythatthenumberofelementsof every given order divides the order of a whole group.They referred to these groupsashavingperfectordersubsets(POS). By using techniques in elementary number theory and group theory, I investigated non-abelian groups, particularly semi-direct products with per- fect order subsets. I wanted to focus on the semi-direct products because a fair amount of research has been done on abelian POS groups. Also, S is an 3 easynon-abelianexampleofagroupwithperfectordersubsets,andS canbe 3 Z (cid:111)Z expressedintheformofasemi-directproduct: . 3 2 Whensemi-directproductsofgroupsareinvolved,itisalwaysnecessary to consider the action. Most of this paper assumes the action is inversion. I then further explore semi-direct products where the action is not inversion. While it is not unusual for POS to occur, it is suprising to see how many dif- ferent categories of groups can be built up by only using a small number of 1 primecomponents. The introductory sections will mainly discuss Finch and Jones’s orginal deﬁnitionsandsomesimpleexamplesaswellasnon-examples. 1.1 METHODOLOGY In the beginning stage of this research, I used GAP to list all non-abelian groupswithperfectordersubsetswhoseordersarelessthan250. Thisallowed me to look for patterns, make conjectures, and create a big picture of formats of POS groups. The GAP code I used is included in the last chapter of this paper. Most of the groups with perfect order subsets resulting from the GAP code are categorized and all investigation is proven in detail. References of resultsthatIusedinthisresearchareclearlyprovided. 1.2 NOTATIONS AND NEWLY INTRODUCED DEF- INITIONS Throughoutthispaper,allgroupsareﬁnite,andforagroupG,wedenote |G| tobetheorderof G and o(x) tobetheorderofagroupelement x in G. As in [1], the order subset of G determined by an element x ∈ G is deﬁned to be thesetOS(x) = {y ∈ G|o(y) = o(x)}. The group G is said to have perfect order subsets (in short, G is called a POS-group) if |OS(x)| is a divisor of |G| for all x ∈ G. We use the standard notation Z to denote the cyclic group of order n with elements 0,1,...,n−1 n underaddition. 2 1.3 BASIC EXAMPLES AND NON-EXAMPLES Example 1.3.1. Let G = Z ×Z ×Z . It is easy to see |G| = 24. We can frame 2 4 3 thefollowingtable: ElementOrder CardinalityofOrderSubset 1 1 2 3 3 2 4 4 6 6 12 8 Every element in the right column, i.e. the number of elements of every given order a divides |G|. Therefore, G hasperfectordersubsets. The following is a short list of some abelian groups that have perfect order subsetsthatwereproveninFinchandJones’paper[1]: • Z2n forall n • (Z )3×Z ×Z 2 3 7 • (Z )4×Z ×Z 2 3 5 • (Z )5×Z ×Z ×Z 2 3 5 31 • (Z )16×Z ×Z ×Z ×Z 2 3 5 17 257 Notonlyabeliangroupshaveperfectordersubsets,butnon-abeliangroups canalsobePOSgroups. Example1.3.2. Let G ∼= S ×Z ×Z .Then |G| = 84,and: 3 2 7 3 ElementOrder CardinatliyofOrderSubset 1 1 2 7 3 2 6 2 7 6 14 42 21 12 42 12 Fromthetable,weknow G hasperfectordersubsets. Here are more examples of non-abelian POS groups that appear in Finch andJones’paper[2]: • S ×(Z )2×Z 3 2 5 • SL(2,q),whereq ∈ {2,3,5,7,11,17,19,41,49,127,251}andwhereSL(2,q) denotesthegroupofall2×2matriceswithdeterminantoneandentries fromtheﬁniteﬁeldF of q elements,where q = pn forsomeprime p. q However, it is not difﬁcult to ﬁnd examples of groups not having perfect Z ordersubsets. Consider ;itisnotaPOSgroupasithas6elementsoforder 7 7. As noticed by Tuan and Hai in [11] and Das in [12], groups that do not haveperfectordersubsetsinclude: • thesymmetricgroup S ,when n ≥ 4 n • D ,when n isaneveninteger. 2n • non-cyclic2-groups 4

ABSTRACT The purpose of this paper is to explore non-abelian ﬁnite groups with per-fect order subsets. A ﬁnite groups is said to have perfect order subsets (POS) if

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