Applications of the Artin-Hasse exponential series and its generalizations to finite algebra groups

67 Pages·2011·0.655 MB·English

by  Darci L. Kracht

APPLICATIONSOFTHEARTIN-HASSEEXPONENTIALSERIES
ANDITSGENERALIZATIONSTOFINITEALGEBRAGROUPS
Adissertationsubmittedto
KentStateUniversityinpartial
fulfillmentoftherequirementsforthe
degreeofDoctorofPhilosophy
by
DarciL.Kracht
December,2011
Dissertationwrittenby
DarciL.Kracht
B.S.,KentStateUniversity,1985
M.A.,KentStateUniversity,1989
Ph.D.,KentStateUniversity,2011
Approvedby
StephenM.Gagola,Jr., Chair,DoctoralDissertationCommittee
DonaldL.White, Member,DoctoralDissertationCommittee
MarkL.Lewis, Member,DoctoralDissertationCommittee
PaulFarrell, Member,DoctoralDissertationCommittee
PeterTandy, Member,DoctoralDissertationCommittee
Acceptedby
AndrewM.Tonge, Chair,DepartmentofMathematicalSciences
TimothyMoerland, Dean,CollegeofArtsandSciences
ii
TABLE OF CONTENTS
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 IntroductionandMotivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 WittVectorsandtheArtin-HasseExponentialSeries . . . . . . . . . . . . . . . . . 6
2.1 Wittvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 TheArtin-Hasseexponentialseries . . . . . . . . . . . . . . . . . . . . . . . . 18
3 SubgroupsDefinedbytheArtin-HasseExponentialSeries. . . . . . . . . . . . . . 27
3.1 Theset E pFxqandthesubgroupEFpxq . . . . . . . . . . . . . . . . . . . . . 27
p p
3.2 Artin-Hasse-exponentiallyclosedsubgroups . . . . . . . . . . . . . . . . . . 32
4 NormalizersofAlgebraSubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1 When Jp (cid:16) 0: theexponentialseries . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 When Jp (cid:24) 0: theArtin-Hasseexponentialseries . . . . . . . . . . . . . . . . 39
5 StrongSubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1 Twocounter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Apower-seriesdescriptionofstrongsubgroups . . . . . . . . . . . . . . . . 49
5.3 Aspecialcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
iii
ACKNOWLEDGMENTS
Itisagreatpleasuretoacknowledgesomeofthemanypeoplewhohavecontributedto
thisdissertation. First,Ioweahugedebtofgratitudetomyadvisor,SteveGagola. Steve
hasaremarkabledepthandbreadthofmathematicalknowledgeandintuition. Iamvery
lucky he has shared this with me over many years. His undergraduate modern algebra
sequence,graduateabstractalgebrasequence,andadvancedgrouptheorycoursesarewhat
led me to pursue graduate work in finite group theory. I thank Steve for his unflagging
supportandpatience,evenwhenItookalonghiatusfromresearch.
IthankDonWhiteforteachingmyfirstcourseincharactertheory,writinganunparal-
leledletterofrecommendation,mixingonehellofaCaucasian,andespeciallyforkeeping
hisHattoriHanzosheathedduringmycandidacyexamanddissertationdefense. Ithank
MarkLewisforadviceandsupport,especiallywhenIfirstbeganmyresearchonalgebra
groups. IthankChuckGartlandforhiscompanyinthedepartmentonmanylateevenings
andweekends,sharingsomegreatbottlesofwine,andforinstallingsomebadlyneeded
font files on my computer— I could not have typeset the big box plus in Theorem 2.13
withoutChuck’shelp.
IthankJoeDiestelformyfirstglimpseintotheworldofmathematicsresearchduringthe
1984BanachSpaceConferenceandforhisinfectiousenthusiasmforallthingsmathematical.
I thank Andrew Tonge for being a great teacher, housemate, friend, and boss. I thank
ArtemZvavitchforgreasingthewheelsofbureaucracyandformanagingtoreconstruct
mylostfilewhenIdecidedtoreturntothedoctoralprogramaftermanyyearsoff. Ithank
UlrikeVorhauerforaSundaymorningspenthelpingmetotranslateandunderstandan
oldGermannumbertheorypaper.
IthankCarlStitzforthemovietrailer. IthankCarolSteinerforheradvice(“Don’tdoit!
iv
Didn’tyoulearnanythingfromwatchingme?”). IthankDanLewis,TerriPolanski,Laura
Dykes, Bev Reed, and many other fellow graduate students, colleagues, and Facebook
friendsfortheirencouragementandsupport.
I thank Dima Ryabogin for his unconventional encouragement (“I don’t understand
whyyounevergotaPh.D.Alltheseotherstupidpeopledoit.”) andfrankcriticismofmy
dissertationdefense(“ThemiddlepartwasOK.”). IthankVolkerMehrmannforclarifying
theshadesofmeaninginthevariousGermanwordsfor“shift”andfortheregularweather
reports from Berlin. I thank Angie Spalsbury for being my supporter, confidante, and
dearest friend as we have traveled the road from student to mathematician and mother
together.
Ithankmyfather,RonKracht,forposinglogicpuzzlesatthedinnertableand,when
hehelpedmewithmymathhomework,foralwaysmakingmeexplainthesolutionsback
tohim. Ithankmymother,LynneKracht,fortheConeofUncertaintyandforproviding
thebestrolemodelaswife,mother,andprofessionalwoman. ItisonlynowthatIrealize
howdifficultitmusthavebeenforhertoreturntocollegeandstartanewcareerwiththree
school-agechildren. Myparentsbelievedinmeandtaughtmetobelieveinmyself.
Finally, I thank my husband, Michael Stacey, and daughter, Claire Stacey, for their
limitlesspatience,love,andsupport,especiallyoverthepasttwoyears,whenIhaveoften
beenabsent,distracted,orshortoftemper. Themanysacrificestheyhavemadeenabledme
topersevereduringthemostdifficulttimes. Theysufferedwithmethrougheveryset-back,
andrejoicedwithmeateverysuccess. IcannotexpresshowgratefulIamtobeabletoshare
thisachievementwiththem.
v
CHAPTER1
INTRODUCTION AND MOTIVATION
Inthischapter,weestablishthecontextforthemainresultsofthisdissertation. Let Fbe
afieldofcharacteristic pandorderq. Let Rbeafinite-dimensionalassociative F-algebra
and J (cid:16) JpRqtheJacobsonradicalof R. (So J isafinite-dimensional,nilpotent,associative
F-algebra.) IfG (cid:16) 1(cid:0)J,thenGisafinite p-group. Groupsofthisformarecalled F-algebra
groups,orsimply,algebragroups. (Wewillassumethisnotationthroughoutthischapter.)
A family of examples is provided by the upper-triangular n(cid:2)n matrices over F, for
a fixed positive integer n. The Jacobson radical, J, of the algebra of all such matrices is
thesetofstrictlyupper-triangularmatricesover F. Thus,G (cid:16) 1(cid:0)J isthesetofunipotent
upper-triangularmatrices,thosewith1’salongthemaindiagonal,oftendenotedUT pFq.
n
ThesubgroupsofG (cid:16) 1(cid:0)J areoftheform1(cid:0)X,where X isasubsetof J closedunder
theoperation
px,yq (cid:222)(cid:209) x(cid:0)y(cid:0)xy. (1.1)
Inparticular, X neednotbeanalgebra. Ontheotherhand,if Lisasubalgebraof J,then L
iscertainlyclosedunder(1.1). Inthiscase,wecall1(cid:0)LanalgebrasubgroupofG. If H ⁄ G
suchthat|HXK|isaq-powerforallalgebrasubgroupsK ofG,thenwesay H isastrong
subgroupofG.
Since algebra subgroups have q-power order and since the collection of algebra sub-
groups of G is closed under intersection, it follows that algebra subgroups are strong.
1
2
However,theconverseisnottrue. Forexample,thesubgroup
1 α α
2
$(cid:4) (cid:12)(cid:7) ,
H (cid:16) ’’ 0 1 (cid:0)α(cid:8) (cid:7)(cid:7)α P F//
’ /
’(cid:6) (cid:14)(cid:7) /
’ /
&(cid:6) (cid:14)(cid:7) .
(cid:6)0 0 1 (cid:14)(cid:7)
(cid:6) (cid:14)(cid:7)
’(cid:6) (cid:14)(cid:7) /
’ /
’’(cid:5) (cid:13)(cid:7) //
’ (cid:7) /
isastrongsubgroupbutnotanalg%ebrasubgroup(cid:7)ofthe-algebragroupUT3pFqfor p ¡ 2.
Here α (cid:16) αpα(cid:1)1q and,moregenerally,forkanintegerwith0 (cid:160) k (cid:160) pandα P F,
2 2
(cid:0) (cid:8)
α αpα(cid:1)1q(cid:4)(cid:4)(cid:4)pα(cid:1)k(cid:0)1q
(cid:16)
k k!
(cid:2) (cid:10)
denotesthegeneralizedbinomialcoefficient.
AlgebragroupsandrelatednotionswereintroducedbyIsaacs(1995)tosolvealong-
standing problem about the character degrees(cid:6) of the upper-triangular matrix groups.
StrongsubgroupsplayedakeyroleinIsaacs’sproofofthatresult.
Inthesamepaper,Isaacsshowedthatcertaininertiasubgroupsofcharactersofalgebra
groups are strong (Theorems 6.1 and 8.3). An inertia subgroup is the stabilizer of a point
undertheactionofthegroupbyconjugationonthesetofirreduciblecharactersofanormal
subgroup. Ofcourse,agroupGalsoactsonitssubgroupsbyconjugation. Recallthatthe
stabilizer of a subgroup H under this action is known as the normalizer of H, denoted
N pHq. Thatis,
G
N pHq (cid:16) tg P G : Hg (cid:16) Hu.
G
InlightoftheresultsofIsaacsaboutcharacterstabilizers,itisnaturaltoaskifnormalizers
of algebra subgroups of algebra groups are strong. This is the question considered in
Chapter4. Thetoolsusedtoanswerthequestionwillbepowerseries.
Thefirstoftheseistheexponentialseries. Ifx P Jwithxp (cid:16) 0,thentheusualexponential
(cid:6)Acharacterisatypeoffunctionfromagrouptoafield,inthiscase,thefieldCofcomplexnumbers. A
characterdegreeisthevalueofacertaintypeofcharacter,calledirreducible,attheidentityofthegroup.Thereis
anextensiveliteratureongroupcharactersandcharacterdegrees.
3
series
x2 x3
expx (cid:16) 1(cid:0)x(cid:0) (cid:0) (cid:0)(cid:4)(cid:4)(cid:4)
2! 3!
isdefined,sinceitterminatesbeforeanydivisionbyzero. Infact,if Jp (cid:16) 0,expdefinesa
bijection J (cid:209) 1(cid:0)J withinverselog: 1(cid:0)J (cid:209) J givenbytheMercatorseries
x2 x3
logp1(cid:0)xq (cid:16) x(cid:1) (cid:0) (cid:1)(cid:4)(cid:4)(cid:4) ,
2 3
for x P J.
For x P J with xp (cid:16) 0 andα P F, define p1(cid:0)xqα (cid:16) exppαlogp1(cid:0)xqq P G, or equiva-
lently,
α α
p1(cid:0)xqα (cid:16) 1(cid:0)αx(cid:0) x2(cid:0) x3(cid:0)(cid:4)(cid:4)(cid:4) .
2 3
(cid:2) (cid:10) (cid:2) (cid:10)
Ifwedefinep1(cid:0)xqF by
p1(cid:0)xqF (cid:16) tp1(cid:0)xqα: α P Fu,
thenp1(cid:0)xqF isasubgroupofG,calledan F-exponentsubgroup. Moreistrue.
Proposition1.1. Ifx P Jwithxp (cid:16) 0,thenp1(cid:0)xqF isasubgroupofGcontaining1(cid:0)x. Ifx (cid:24) 0,
thenp1(cid:0)xqF isisomorphictotheadditivegroupof F. Moreover,distinct F-exponentsubgroups
intersecttrivially. Finally, everyalgebrasubgroupof G ofexponent p istheunionofitsdistinct
F-exponentsubgroups.
Proof. ThisisCorollary5.2of(Isaacs,1995).
Notice that if we define exppFxˆq (cid:16) texppαxˆq: α P Fu, then p1(cid:0)xqF (cid:16) exppFxˆq for
xˆ (cid:16) logp1(cid:0)xq, so we may denote F-exponent subgroups by p1(cid:0)xqF or by exppFxˆq, as
convenient.
Let H beasubgroupofGofexponent p. Wesay H isexponentiallyclosedifexppFxq (cid:132) H
wheneverexppxq P H. Isaacs(1995)calledthesesubgroupspartitionedsubgroupsbecause
theyareequallypartitionedinthesensedefinedin(Isaacs,1973). ItfollowsfromProposition
4
1.1 that F-exponent subgroups and algebra subgroups of exponent p are exponentially
closedandthattheintersectionofexponentiallyclosedsubgroupsisitselfexponentially
closed.
Thenextresultiscentralto§4.1,wherewegiveanaffirmativeanswertothequestion
aboutnormalizersinthecasewhere Jp (cid:16) 0.
Proposition1.2. IfGisanalgebragroup,theneveryexponentiallyclosedsubgroupofGisstrong.
Proof. ThisisLemma5.3of(Isaacs,1995).
Thenotionsof F-exponentandexponentiallyclosedsubgroupsareoflimiteduseinan
algebragroupwhoseexponentexceedsitscharacteristic. Therefore,wesetouttogeneralize
these ideas using different power series. In fact, other power series have been used in
similarcircumstances. CharacterdegreeresultsanalogoustothoseofIsaacs(1995)butfor
otherclassicalmatrixgroupsweregivenbyPrevitali(1995). Heusedthepowerseries
8 p(cid:1)1qk(cid:1)1
σpxq (cid:16) x(cid:0) 1(cid:0)x2 (cid:16) x(cid:0) C x2k,
22k(cid:1)1 k(cid:1)1
k(cid:16)0
a ‚
whereC (cid:16) 1 2n P Z: isthen-thCatalannumberforn ¥ 0andC (cid:16) (cid:1)1. Thisseries
n n(cid:0)1 n (cid:1)1 2
(cid:0) (cid:8)
is defined in odd characteristic only, but these results were known to be false for p (cid:16) 2,
anyway. Previtaliobtainedhisresultsaboutcharacterdegreesbyusingthispowerseries
toshowthatcertainsectionsofthegroupsarestrong. Previtali(1999)laterusedtruncated
exponentialseriestoprovesimilarresultsforsizesofcertainconjugacyclassesinclassical
groups.
Forourpurposes,wewantedapowerseries,orfamilyofpowerseries,thatcouldbe
usedinanycharacteristicandinalgebragroupsofarbitrarilylargeexponent. Thetruncated
exponentialseriesdonothavethepropertiesweneed. Fortunately,thereisageneralization
oftheexponentialseriescalledtheArtin-Hasseexponentialseriesthatsuitsourpurpose,
:Asusual,ZdenotesthesetofallintegersandQdenotesthesetofallrationalnumbers.
5
foundinthenumbertheoryliterature. InChapter2,weprovidethebackgroundmaterial
neededtoprovethattheArtin-Hasseexponentialserieshasthedesiredproperties. This
includessomeelementaryresultsabouttheringofWittvectors. InChapter3,wedevelop
analogsof F-exponentsubgroupsandexponentiallyclosedsubgroupsfortheArtin-Hasse
exponentialseries. InChapter4,weusethesetoolstoshowwhennormalizersofalgebra
subgroupsarestrongandwhentherearecounter-examples.
Aswehaveseen,strongsubgroupsplayanimportantroleintheresultsofIsaacs(1995)
andPrevitali(1995)aboutcharacterdegreesinclassicalgroups. Morerecently,Andre´ (2010)
hasobtainedsimilarresultsinamoregeneralsettingbyshowingthatcertainfixed-point
subgroupsarestrong. TheseresultsaswellasourresultsinChapter4motivatethestudy
ofstrongsubgroupsinChapter5. Hereagain,weusepowerseries,butthistimetheseries
canbemuchmoregeneral.