Non-Abelian Groups with Perfect Order Subsets

46 Pages·2014·0.29 MB·English

by  Hongying Zhao

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Mathematics Student Work Mathematics
2012
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Hongying Zhao
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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