Noether's problem for abelian extensionsof cyclic p-groups

26 Pages·2014·0.4 MB·English

by  Ivo M. Michailov

Pacific
Journal of
Mathematics
NOETHER’S PROBLEM FOR ABELIAN EXTENSIONS
OF CYCLIC p-GROUPS
IVO M. MICHAILOV
Volume270 No.1 July2014
PACIFICJOURNALOFMATHEMATICS
Vol.270,No.1,2014
dx.doi.org/10.2140/pjm.2014.270.167
NOETHER’S PROBLEM FOR ABELIAN EXTENSIONS
OF CYCLIC p-GROUPS
IVO M. MICHAILOV
Inlovingmemoryofmydearmother
Let K beafieldandGafinitegroup. LetGactontherationalfunctionfield
K(x(g):g∈G)by K-automorphismsdefinedby g·x(h)=x(gh)forany g,
h∈G. Denoteby K(G)thefixedfield K(x(g):g∈G)G. Noether’sproblem
thenaskswhether K(G)isrational(i.e.,purelytranscendental)over K. The
firstmainresultofthisarticleisthat K(G)isrationalover K foracertain
classof p-groupshavinganabeliansubgroupofindex p. Thesecondmain
resultisthat K(G)isrationalover K foranygroupoforder p5or p6(where
pisanoddprime)havinganabeliannormalsubgroupsuchthatitsquotient
group is cyclic. (In both theorems we assume that if charK (cid:54)= p then K
containsaprimitive pe-throotofunity,where pe istheexponentofG.)
1. Introduction
Let K beafield. Afieldextension L of K iscalledrationalover K (or K-rational,
for short) if L (cid:39) K(x ,...,x ) for some integer n, with x ,...,x algebraically
1 n 1 n
independent over K. Now let G be a finite group. Let G act on the rational
functionfield K(x(g):g∈G)by K-automorphismsdefinedby g·x(h)=x(gh)
for any g,h ∈ G. Denote by K(G) the fixed field K(x(g): g ∈ G)G. Noether’s
problemthenaskswhether K(G)isrationalover K. Thisisrelatedtotheinverse
Galoisproblem,totheexistenceofgeneric G-Galoisextensionsover K,andtothe
existenceofversal G-torsorsover K-rationalfieldextensions[Swan1983;Saltman
1982;Garibaldietal.2003,§33.1,p.86]. Noether’sproblemforabeliangroupswas
studiedextensivelybySwan,Voskresenskii,Endo,MiyataandLenstra,etc. The
readerisreferredto[Swan1983]forasurveyofthisproblem. Fischer’stheoremisa
startingpointofinvestigatingNoether’sproblemforfiniteabeliangroupsingeneral.
Theorem1.1(Fischer[Swan1983,Theorem6.1]). LetG beafiniteabeliangroup
ofexponente. Assumethat(i)eithercharK =0orcharK >0withcharK (cid:45)e,and
(ii) K containsaprimitivee-throotofunity. Then K(G)isrationalover K.
MSC2010: primary14E08,14M20;secondary13A50,12F12.
Keywords: Noether’sproblem,rationalityproblem,metabeliangroupactions.
167
168 IVOM.MICHAILOV
Onthe otherhand, justahandful ofresultsabout Noether’sproblemhavebeen
obtainedwhenthegroups arenonabelian. Thisisthecaseevenwhenthe group G
isa p-group. Thereaderisreferredto[ChuandKang2001;HuandKang2010;
Kang2006;2011;2009]forpreviousresultsonNoether’sproblemfor p-groups.
ThefollowingtheoremofKanggeneralizesFischer’stheoremforthemetacyclic
p-groups.
Theorem 1.2 [Kang 2006, Theorem 1.5]. Let G be a metacyclic p-group with
exponent pe,andlet K beanyfieldsuchthat(i)charK = p,or(ii)charK (cid:54)= pand
K containsaprimitive pe-throotofunity. Then K(G)isrationalover K.
ThenextjobistostudyNoether’sproblemformetabeliangroups. Threeresults
duetoHaeuslein,HajjaandKang,respectively,areknown.
Theorem1.3[Haeuslein1971]. Let K beafieldandG beafinitegroup. Assume
that(i)G containsanabeliannormalsubgroup H suchthatG/H iscyclicofprime
order p,(ii)(cid:90)[ζ ]isauniquefactorizationdomain,and(iii)ζ ∈ K,wheree is
p pe
theexponentofG. IfG →GL(V)isanyfinite-dimensionallinearrepresentation
ofG over K,then K(V)G isrationalover K.
Theorem1.4[Hajja1983]. Let K beafieldandG beafinitegroup. Assumethat
(i)G containsanabeliannormalsubgroup H suchthatG/H iscyclicofordern,
(ii)(cid:90)[ζ ]isauniquefactorizationdomain,and(iii) K isalgebraicallyclosedwith
n
charK =0. If G →GL(V) is any finite-dimensional linear representation of G
over K,then K(V)G isrationalover K.
Theorem1.5[Kang2009,Theorem1.4]. Let K beafieldandG beafinitegroup.
Assumethat(i)G containsanabeliannormalsubgroup H suchthatG/H iscyclic
ofordern,(ii)(cid:90)[ζ ]isauniquefactorizationdomain,and(iii)ζ ∈ K,whereeis
n e
theexponentofG. IfG →GL(V)isanyfinite-dimensionallinearrepresentation
ofG over K,then K(V)G isrationalover K.
Notethatthoseintegersn forwhich(cid:90)[ζ ]isauniquefactorizationdomainare
n
determinedbyMasleyandMontgomery.
Theorem 1.6 [Masley and Montgomery 1976]. (cid:90)[ζ ] is a unique factorization
n
domainifandonlyif 1≤n≤22,orn=24,25,26,27,28,30,32,33,34,35,36,
38,40,42,45,48,50,54,60,66,70,84,90.
Therefore,Theorem1.3holdsonlyforprimes p suchthat1≤ p≤19. Oneof
thegoalsofourpaperistoshowthatthethisconditioncanbewaived,undersome
additionalassumptionsregardingthestructureoftheabeliansubgroup H.
Considerthefollowingsituation. LetG beagroupoforder pn forn≥2withan
abeliansubgroup H oforder pn−1. Bender[1927/28]determinedsomeinteresting
propertiesofthesegroups. Westudyfurtherthecasewhenthe p-thlowercentral
NOETHER’SPROBLEMFORABELIANEXTENSIONSOFCYCLIC p-GROUPS 169
subgroup G(p) is trivial. (Recall that G(0) = G and G(i) =[G,G(i−1)] for i ≥1
form the so-called lower centralseries.) Forour purposes we needto classify with
generatorsandrelationsthesegroups. Weachievethisinthefollowinglemma.
Lemma1.7. LetG beagroupoforder pn forn≥2withanabeliansubgroup H of
order pn−1. Chooseanyα∈G suchthatα generatesG/H,thatis,α∈/ H,αp∈ H.
Define H(p)={h∈H :hp=1,h∈/ Hp}∪{1},andassumethat[H(p),α]⊂H(p).
Assumealsothatthe p-thlowercentralsubgroupG(p) istrivial. Then H isadirect
productofnormalsubgroupsofG belongingtofourtypes:
(1) (C )s for some s ≥1. There exist generators α ,...,α of (C )s such that
p 1 s p
[αj,α]=αj+1 for1≤ j ≤s−1andαs ∈ Z(G).
(2) C forsomea≥1. Thereexistsageneratorβ of C suchthat[β,α]=βbpa−1
pa pa
forsomeb:0≤b≤ p−1.
(3) Cpa1 ×Cpa2 ×···×Cpak ×(Cp)s for some k ≥ 1, ai ≥ 2, s ≥ 1. There
exist generators α11, α21, ...,αk1 of Cpa1 × Cpa2 × ··· × Cpak such that
[αi,1,α]=αip+ai1+,11−1 ∈ Z(G)fori =1,...,k−1. Therealsoexistgenerators
αk,2,...,αk,s+1 of (Cp)s such that [αk,j,α] = αk,j+1 for 1 ≤ j ≤ s and
αk,s+1∈ Z(G).
(4) Cpa1 ×Cpa2 ×···×Cpak forsomek ≥2,ai ≥2. Foranyi :1≤i ≤k there
existsageneratorαi,1 ofthefactorCpai suchthat[αi,1,α]=αip+ai1−,11 ∈ Z(G)
and [αk,1,α]∈(cid:10)α1p,a11−1,...,αkp,a1k−1(cid:11).
ThefirstmainresultofthispaperisageneralizationofTheorem1.3:
Theorem1.8. LetG beagroupoforder pn forn≥2withanabeliansubgroup H
oforder pn−1,andletG beofexponent pe. Chooseanyα∈G suchthatαgenerates
G/H,thatis,α∈/ H,αp ∈ H. Define H(p)={h∈ H :hp =1,h∈/ Hp}∪{1},and
assumethat[H(p),α]⊂H(p). Letthe p-thlowercentralsubgroupG(p) betrivial.
Assume that (i) charK = p > 0, or (ii) charK (cid:54)= p and K contains a primitive
pe-throotofunity. Then K(G)isrationalover K.
The key idea to prove Theorem 1.8 is to find a faithful G-subspace W of the
regularrepresentationspace(cid:76) K·x(g)andtoshowthatWG isrationalover K.
g∈G
The subspace W is obtained as an induced representation from H by applying
Lemma1.7.
The next goal of our article is to study Noether’s problem for some groups of
orders p5 and p6 foranyoddprime p. Weusethelistofgeneratorsandrelations
forthesegroups,givenbyJames[1980]. Itisknownthat K(G)isalwaysrational
if G is a p-group of order at most p4 and ζ ∈ K, where e is the exponent of G
e
170 IVOM.MICHAILOV
(see[ChuandKang2001]). However,in[HoshiandKang2011]itisshownthat
thereexistsagroup G oforder p5 suchthat(cid:67)(G)isnotrationalover(cid:67).
The second main result of this article is the following rationality criterion for
the groups of orders p5 and p6 having an abelian normal subgroup such that its
quotientgroupiscyclic.
Theorem 1.9. Let G be a group of order pn for n ≤ 6 with an abelian normal
subgroup H such that G/H is cyclic. Let G be of exponent pe. Assume that
(i) charK = p >0, or (ii) charK (cid:54)= p and K contains a primitive pe-th root of
unity. Then K(G)isrationalover K.
WedonotknowwhetherTheorem1.9holdsforanyn≥7. However,weshould
not“overgeneralize”Theorem1.9tothecaseofanymetabeliangroupbecauseof
thefollowingtheoremofSaltman.
Theorem1.10[Saltman1984]. Foranyprimenumber p andforanyfield K with
charK (cid:54)= p (in particular, K may be an algebraically closed field), there is a
metabelian p-groupG oforder p9 suchthat K(G)isnotrationalover K.
We organize this paper as follows. We recall some preliminaries in Section 2
that will be used in the proofs of Theorems 1.8 and 1.9. There we also prove
Lemma 2.5, which is a generalization of Kang’s argument [2011, Case 5, Step
II]. In Section 3 we prove Lemma 1.7, which is of independent interest, since it
provides a list of generators and relations for any p-group G having an abelian
subgroup H ofindex p,provided that[H(p),α]⊂ H(p)and G(p)=1. Our main
results—Theorems1.8and1.9—areprovedinSections4and5,respectively.
2. Preliminaries
Welistseveralresultswhichwillbeusedinthesequel.
Theorem2.1[HajjaandKang1995,Theorem1]. LetG beafinitegroupactingon
L(x ,...,x ),therationalfunctionfieldofm variablesoverafield L,suchthat
1 m
(1) foranyσ ∈G,σ(L)⊂ L,
(2) therestrictionoftheactionofG to L isfaithful,
(3) foranyσ ∈G,
σ(x ) x 
1 1
. .
 .. = A(σ) .. +B(σ),
   
σ(x ) x
m m
where A(σ)∈GL (L)and B(σ)isanm×1matrixover L. Thenthereexist
m
z ,...,z ∈ L(x ,...,x )suchthat L(x ,...,x )G = LG(z ,...,z )and
1 m 1 m 1 m 1 m
σ(z )=z foranyσ ∈G and1≤i ≤m.
i i
NOETHER’SPROBLEMFORABELIANEXTENSIONSOFCYCLIC p-GROUPS 171
Theorem2.2[Ahmadetal.2000,Theorem3.1]. LetG beafinitegroupactingon
L(x), the rational function field of one variable over a field L. Assume that, for
anyσ ∈G,σ(L)⊂ L andσ(x)=aσx+bσ foranyaσ,bσ ∈ L withaσ (cid:54)=0. Then
L(x)G = LG(z)forsomez∈ L[x].
Theorem2.3[Chu and Kang 2001, Theorem 1.7]. IfcharK = p>0and G isa
finite p-group,then K(G)isrationalover K.
Thefollowinglemmacanbeextractedfromsomeproofsin[Kang2011;Huand
Kang2010].
Lemma2.4. Let(cid:104)τ(cid:105)beacyclicgroupofordern>1,actingon K(v1,...,vn−1),
therationalfunctionfieldof n−1variablesoverafield K,suchthat
τ :v (cid:55)→v (cid:55)→···(cid:55)→v (cid:55)→(v ···v )−1(cid:55)→v .
1 2 n−1 1 n−1 1
Supposethat K containsaprimitiven-throotofunityξ. Then K(v ,...,v )=
1 n−1
K(s1,...,sn−1),whereτ :si (cid:55)→ξisi for1≤i ≤n−1.
Proof. Definew0=1+v1+v1v2+···+v1v2···vn−1,w1=(1/w0)−1/n,wi+1=
(v1v2···vi/w0)−1/n for1≤i ≤n−1. Thus K(v1,...,vn−1)=K(w1,...,wn)
withw +w +···+w =0and
1 2 n
τ :w (cid:55)→w (cid:55)→···(cid:55)→w (cid:55)→w (cid:55)→w .
1 2 n−1 n 1
Defines =(cid:80) ξ−ijw for1≤i ≤n−1. Thenτ :s (cid:55)→ξis for1≤i ≤n−1
i 1≤j≤n j i i
and K(w1,...,wn)= K(s1,...,sn−1). (cid:3)
Next,generalizinganargumentusedin[Kang2011,Case5,StepII],weobtain
aresultthatwillplayanimportantroleinourwork.
Lemma2.5. Letk >1,let p beanyprimeandlet(cid:104)α(cid:105)beacyclicgroupoforder p,
actingon K(y ,y ,...,y :1≤i ≤ p−1),therationalfunctionfieldofk(p−1)
1i 2i ki
variablesoverafield K,suchthat
α: yj1(cid:55)→ yj2(cid:55)→···(cid:55)→ yjp−1(cid:55)→(yj1yj2···yjp−1)−1 for1≤ j ≤k.
AssumethatK(v ,v ,...,v :1≤i≤p−1)=K(y ,y ,...,y :1≤i≤p−1)
1i 2i ki 1i 2i ki
where for any j : 1 ≤ j ≤ k and for any i : 1 ≤ i ≤ p−1 the variable v is a
ji
monomial in the variables y ,y ,...,y . Assume also that the action of α on
1i 2i ki
K(v ,v ,...,v :1≤i ≤ p−1)isgivenby
1i 2i ki
α:v (cid:55)→v vp , v (cid:55)→v (cid:55)→···(cid:55)→v (cid:55)→ A (v vp−1vp−2···v2 )−1
j1 j1 j2 j2 j3 jp−1 j j1 j2 j3 jp−1
for 1 ≤ j ≤ k, where Aj is some monomial in v1i,...,vj−1i for 2 ≤ j ≤ k and
A =1. If K containsaprimitive p-throotofunityζ,then
1
K(v ,v ,...,v :1≤i ≤ p−1)= K(s ,s ,...,s :1≤i ≤ p−1),
1i 2i ki 1i 2i ki
172 IVOM.MICHAILOV
whereα:s (cid:55)→ζis for1≤ j ≤k,1≤i ≤ p−1.
ji ji
Proof. We write the additive version of the multiplication action of α; that is,
considerthe(cid:90)[π]-moduleM=(cid:76) (cid:0)(cid:76) (cid:90)·v (cid:1),whereπ=(cid:104)α(cid:105). Define
1≤m≤k 1≤i≤p−1 mi
submodules M = (cid:76) (cid:0)(cid:76) (cid:90)·v (cid:1) for 1 ≤ j ≤ k. Thus α has the
j 1≤m≤j 1≤i≤p−1 mi
followingadditiveaction
α:v (cid:55)→v +pv ,
j1 j1 j2
vj2(cid:55)→vj3(cid:55)→···(cid:55)→vjp−1(cid:55)→Aj−vj1−(p−1)vj2−(p−2)vj3−···−2vjp−1,
where Aj ∈ Mj−1.
By Lemma 2.4, M is isomorphic to the (cid:90)[π]-module N = (cid:76) (cid:90)·u ,
1 1≤i≤p−1 i
whereu =v ,u =αi−1·v for2≤i ≤ p−1,and
1 12 i 12
α:u (cid:55)→u (cid:55)→···(cid:55)→u (cid:55)→−u −u −···−u (cid:55)→u .
1 2 p−1 1 2 p−1 1
Let(cid:56) (T)∈(cid:90)[T]bethe p-thcyclotomicpolynomial. Since(cid:90)[π]isisomorphic
p
to (cid:90)[T]/(Tp−1), we find that (cid:90)[π]/(cid:56) (α)(cid:39)(cid:90)[T]/(cid:56) (T)(cid:39)(cid:90)[ω], the ring of
p p
p-th cyclotomic integers. As (cid:56) (α)·x =0 for any x ∈ N, the (cid:90)[π]-module N
p
can be regarded as a (cid:90)[ω]-module through the morphism (cid:90)[π]→(cid:90)[π]/(cid:56) (α).
p
When N is regarded as a (cid:90)[ω]-module, we have N (cid:39) (cid:90)[ω], the rank-one free
(cid:90)[ω]-module.
Weclaimthat M itselfcanberegardedasa(cid:90)[ω]-module,thatis,(cid:56) (α)·M =0.
p
Wereturntomultiplicativenotation. Notethatallv aremonomialsinthe y .
ji ji
Theactionofα on y giveninthestatementsatisfies(cid:81) αm(y )=1for
ji 0≤m≤p−1 ji
any1≤ j ≤k,1≤i ≤ p−1. Usingtheadditivenotations,weget(cid:56) (α)·y =0.
p ji
Hence(cid:56) (α)·M =0.
p
Define M(cid:48)= M/Mk−1. Wehaveashortexactsequenceof(cid:90)[π]-modules
(2-1) 0→ Mk−1→ M → M(cid:48)→0.
Since M isa(cid:90)[ω]-module,(2-1)isashortexactsequenceof(cid:90)[ω]-modules. Pro-
ceeding by induction, we obtain that M is a direct sum of free (cid:90)[ω]-modules
isomorphicto N. Hence, M (cid:39)(cid:76) N ,where N (cid:39) N isafree(cid:90)[ω]-module
1≤j≤k j j
andsoa(cid:90)[π]-modulealso(for1≤ j ≤k).
Finally,we interpret theadditiveversionof M (cid:39)(cid:76) N (cid:39) Nk intermsof
1≤j≤k j
themultiplicativeversionasfollows: Thereexistw thataremonomialsinv for
ji ji
1≤ j ≤k,1≤i ≤ p−1suchthat K(w )= K(v )andα actsas
ji ji
α:wj1(cid:55)→wj2(cid:55)→···(cid:55)→wjp−1(cid:55)→(wj1wj2···wjp−1)−1 for1≤ j ≤k.
AccordingtoLemma2.4,theaboveactioncanbelinearizedaspointedoutinthe
statement. (cid:3)
NOETHER’SPROBLEMFORABELIANEXTENSIONSOFCYCLIC p-GROUPS 173
Now,let G beanymetacyclic p-groupgeneratedbytwoelementsσ andτ with
relationsσpa =1,τpb=σpc andτ−1στ =σε+δpr whereε=1if pisodd,ε=±1if
p=2,δ=0,1anda,b,c,r≥0aresubjecttosomerestrictions. Forthedescription
oftheserestrictionssee,forexample,[Kang2006,p.564].
Theorem 2.6 [Kang 2006, Theorem 4.1]. Let p be a prime number, m, n andr
positiveintegers,k =1+ pr if(p,r)(cid:54)=(2,1)ork =−1+2r if p=2andr ≥2.
Let G be a split metacyclic p-group of order pm+n and exponent pe defined by
G =(cid:104)σ,τ :σpm =τpn =1,τ−1στ =σk(cid:105). Let K beanyfieldsuchthatcharK (cid:54)= p
and K contains a primitive pe-th root of unity, and let ζ be a primitive pm-th
root of unity. Then K(x0,x1,...,xpn−1)G is rational over K, where G acts on
x ,...,x by
0 pn−1
σ :x (cid:55)→ζkix , τ :x (cid:55)→x (cid:55)→···(cid:55)→x (cid:55)→x .
i i 0 1 pn−1 0
3. ProofofLemma1.7
Itiswellknownthat H isanormalsubgroupof G. Wedividetheproofintosteps.
StepI. Letβ1 beanyelementof H thatisnotcentral. SinceG(p)={1},thereexist
β2,...,βk∈H forsomek:2≤k≤ psuchthat[βj,α]=βj+1,where1≤ j≤k−1
andβ (cid:54)=1iscentral. Wearegoingtoshownowthattheorderofβ isnotgreater
k 2
than p. In particular, from the multiplication rule [a,α][b,α]=[ab,α] (for any
a,b∈ H)itfollowsthatall p-thpowersarecontainedinthecenterof G.
From[βj,α]=βj+1 therefollowsthewellknownformula
(p) (p) ( p )
(3-1) α−pβ1αp =β1β21 β32 ···βpp−1 βp+1,
whereweputβk+1=···=βp+1=1. Sinceαp isin H,weobtaintheformula
(p) (p) ( p )
β 1 β 2 ···β k−1 =1.
2 3 k
Hence(cid:0)β ·(cid:81) βaj(cid:1)p=1forsomeintegersa . Itisnothardtoseethatthisidentity
2 j(cid:54)=2 j j
isimpossibleiftheorderofβ exceeds p. Indeed,if(cid:96)=max{j:βp(cid:54)=1},thenβp is
2 j (cid:96)
inthesubgroupgeneratedbyβp,...,βp . Thus[βp,α]=[βb2p···βb(cid:96)−1p,α]=
2 (cid:96)−1 (cid:96) 2 (cid:96)−1
β3b2p···β(cid:96)b(cid:96)−1p (cid:54)= 1 for some b2, ..., b(cid:96)−1 ∈ (cid:90)p. On the other hand, [β(cid:96)p,α] =
βp =1,whichisacontradiction.
(cid:96)+1
StepII. Letuswritethedecompositionof H asadirectproductofcyclicsubgroups
(not necessarily normal in G): H (cid:39)(Cp)t ×Cpa1 ×Cpa2 ×···×Cpas for 0≤t,
2≤a1 ≤a2 ≤···≤as. Choose a generator α11 ∈Cpa1. Since G(p) ={1}, there
exist α12,...,α1k ∈ H for some k :2≤k ≤ p such that [α1j,α]=α1j+1, where
1 ≤ j ≤ k −1 and α (cid:54)= 1 is central. From Step I it follows that the order of
1k
α is not greater than p. We are going to define a normal subgroup of G which
12
174 IVOM.MICHAILOV
dependsonthenatureoftheelementα . Wewilldenoteitby(cid:104)(cid:104)α (cid:105)(cid:105),andcallit
12 11
thecommutatorchainof α . Simultaneously,wewilldefineacomplementin H
11
denotedby(cid:104)(cid:104)α (cid:105)(cid:105).
11
CaseII.1. Letα =αpa1−1c forsomec :0≤c ≤ p−1. Define(cid:104)(cid:104)α (cid:105)(cid:105)=(cid:104)α (cid:105)
12 11 1 1 1 11 11
and(cid:104)(cid:104)α (cid:105)(cid:105)=(C )t·(cid:104)α ,...,α (cid:105). Clearly,(cid:104)(cid:104)α (cid:105)(cid:105)isanormalsubgroupoftype2.
11 p 21 s1 11
CaseII.2. Let α ∈/ Hp. According to the assumptions of our lemma, we have
12
[H(p),α]∩ Hp = {1}, so α ∈/ Hp for all j. Define (cid:104)(cid:104)α (cid:105)(cid:105) = (cid:104)α ,...,α (cid:105).
1j 11 11 1k
Then (cid:104)(cid:104)α11(cid:105)(cid:105)(cid:39)Cpa1 ×(Cp)k−1 is a normal subgroup of type 3. Define (cid:104)(cid:104)α11(cid:105)(cid:105)=
(C )t−k+1·(cid:104)α ,...,α (cid:105),where(C )t−k+1isthecomplementof(C )k−1in(C )t.
p 21 s1 p p p
CaseII.3. Let α ∈ Hp. Then α = (cid:81) αpai−1di, where A ⊂ {1,2,...,s},
12 12 i∈A i1
1≤d ≤ p−1. Puti =min{i ∈ A}.
i 0
Ifi0=1,thenα12=(cid:0)α1d11(cid:81)i∈A,i(cid:54)=1αip1ai−a1di(cid:1)pa1−1. Wereplacethegeneratorα11
with α1(cid:48)1 =α1d11(cid:81)i∈A,i(cid:54)=1αip1ai−a1di. Clearly, ordα1(cid:48)1 =ordα11 and [α1(cid:48)1,α]∈(cid:104)α1(cid:48)1(cid:105),
sothiscaseisreducedtoCaseI.
witIhfiα0i(cid:48)0>11=,tαhide0in10 (cid:81)α1i2∈=A,i(cid:0)(cid:54)=αiid00i10α(cid:81)ip1aii−∈aAi0,dii(cid:54)=.i0Cαleip1aari−lyai,0doir(cid:1)dpαai0i(cid:48)0−11.=Woerrdepαlia01ceanthdeαgi(cid:48)e0pn1aie0r−a1to=rααi1021.
Abusing notation we will assume henceforth that i = 2 and αpa2−1 = α .
0 21 12
Considerα =[α ,α]. Wehavethreepossibilitiesnow.
22 21
SubcaseII.3.1. Ifα ∈(cid:104)αpa1−1,αpa1−1(cid:105),define(cid:104)(cid:104)α (cid:105)(cid:105)=(cid:104)α ,α (cid:105). Then(cid:104)(cid:104)α (cid:105)(cid:105)(cid:39)
22 11 21 11 11 21 11
Cpa1 ×Cpa2 isanormalsubgroupoftype4.
SubcaseII.3.2. If α22 ∈/ Hp, there exist α22,...,α2(cid:96) ∈ H for some (cid:96):2≤(cid:96)≤ p
such that [α2j,α] = α2j+1, where 1 ≤ j ≤ (cid:96)−1 and α2(cid:96) (cid:54)= 1 is central. Define
(cid:104)(cid:104)α11(cid:105)(cid:105)=(cid:104)α11,α21,α22,...,α2(cid:96)(cid:105). Then(cid:104)(cid:104)α11(cid:105)(cid:105)(cid:39)Cpa1×Cpa2×(Cp)(cid:96)−1isanormal
subgroupoftype3.
SubcaseII.3.3. α ∈ Hp. Accordingtotheobservationswehavejustmade,this
22
subcaseleadstothefollowingtwofinalpossibilities.
• α22 = α3p1a3−1,...,αr−12 = αrp1ar−1,αr2 ∈ (cid:104)α1p1a1−1,...,αrp1ar−1(cid:105). Define (cid:104)(cid:104)α11(cid:105)(cid:105) =
(cid:104)α11,α21,...,αr1(cid:105). Then(cid:104)(cid:104)α11(cid:105)(cid:105)(cid:39)Cpa1 ×Cpa2 ×···×Cpar isanormalsubgroup
oftype4. Define(cid:104)(cid:104)α11(cid:105)(cid:105)=(Cp)t ·(cid:104)αr+11,...,αs1(cid:105).
• α22 =α3p1a3−1,...,αr−12 =αrp1ar−1,αr2 ∈/ Hp. Then there exist αr2,...,αr(cid:96) ∈ H
for some (cid:96) : 2 ≤ (cid:96) ≤ p such that [αrj,α] = αrj+1, where 1 ≤ j ≤ (cid:96)−1 and
αr(cid:96) (cid:54)=1 is central. Define (cid:104)(cid:104)α11(cid:105)(cid:105)=(cid:104)α11,α21,...,αr1,αr2,...,αr(cid:96)(cid:105). In this case
(cid:104)(cid:104)α11(cid:105)(cid:105) (cid:39) Cpa1 ×Cpa1 ×···×Cpar ×(Cp)(cid:96)−1 is a normal subgroup of type 3.
Define(cid:104)(cid:104)α11(cid:105)(cid:105)=(Cp)t−(cid:96)+1·(cid:104)αr+11,...,αs1(cid:105),where(Cp)t−(cid:96)+1 isthecomplement
of(C )(cid:96)−1 in(C )t.
p p
StepIII. Put H =(cid:104)(cid:104)α (cid:105)(cid:105)and H =(cid:104)(cid:104)α (cid:105)(cid:105). Notethat H ∩H ={1}. However, H
1 11 2 11 1 2 2
may notbe anormal subgroupof G. That iswhy we need to show that there exist
NOETHER’SPROBLEMFORABELIANEXTENSIONSOFCYCLIC p-GROUPS 175
acommutatorchainH andanormalsubgroupH of G suchthat H =H ×H .
1 2 1 2
Inthisstep,wewilldescribeasomewhatalgorithmicapproachwhichreplacesthe
generatorsof H untilthedesiredresultisobtained.
Assume henceforth that H is not normal in G. Then there exists a generator
2
β∈H suchthatα−1βα=hh forsomeh∈H ,h ∈H ,h ∈/ H . Sinceh=βh
2 1 2 1 1 1 2 2
forsome h ∈ H ,weget[β,α]=h h .
2 2 1 2
Letusassumefirstthatordβ = p. If h ∈ Hp,then h ∈/[H(p),α];otherwise
1 2
[H(p),α]∩Hp (cid:54)={1}. Inotherwords, h doesnotappearinsimilarchains,sowe
2
cansimplyput h h ,insteadof h ,asageneratorof H . Inthiswayweobtaina
1 2 2 2
group that is G-isomorphic to H . Thus we get that [β,α] is in this new copy of
2
H . Similarly,if h ∈ H(p)and h ∈/[H(p),α],wecanobtainanewcopyof H
2 1 2 2
such that [β,α] is in H . If h ∈[H(p),α], we may assume that [β,α]∈ H . In
2 2 1
thiscase(cid:104)(cid:104)α11(cid:105)(cid:105)mustbeoftype3. Let(cid:104)(cid:104)α11(cid:105)(cid:105)(cid:39)Cpa1 ×Cpa2 ×···×Cpak ×(Cp)s
begeneratedbyelementsα11,...,αk1,αk2,...,αks+1 withrelationsgiveninthe
statement of the lemma. Assume that αk(cid:96) = [β,α] for some (cid:96) : 2 ≤ (cid:96) ≤ s +1.
If (cid:96) > 2, replace β with β(cid:48) = βα−1 . Hence [β(cid:48),α] = 1. If (cid:96) = 2, we can put
k(cid:96)−1
α(cid:48) =α β−1,insteadofα ,asageneratorof H . Inthiswayweobtainagroupof
k1 k1 k1 1
type4,since[α(cid:48) ,α]=1. Clearly,[β,α]isnotinthisnewcommutatorchainH .
k1 1
It is not hard to see that with similar replacements we can treat the general case
[β,α]=(cid:81)i αip1ai−1ci ·(cid:81)j αkj. Thus we obtain the decomposition H =H1×H2,
whereH andH arenormalsubgroupsof G.
1 2
Next,wearegoingtoassumethatordβ > p. Accordingtothedefinitionofthe
commutatorchainofα weneedtoconsiderthethreecasesofStepIIseparately.
11
CaseIII.1. α12 = α1p1a1−1c1 for some c1 : 1 ≤ c1 ≤ p −1. Here we must have
h1=α1p1a1−1d1 forsomed1:1≤d1≤ p−1. Wecanreplaceβ withβ(cid:48)=βα1−1d1/c1,
so[β(cid:48),α]=h .
2
CaseIII.2. α12 ∈/ Hp. If h1 = (cid:81)j≥2α1djj for some dj : 0 ≤ dj ≤ p−1, we can
replaceβ withβ(cid:48)=β(cid:81) α−dj . Hence[β(cid:48),α]=h . Thisreducestheanalysisto
j≥2 1j−1 2
thecaseh1=α1p1a1−1d1 forsomed1:0≤d1≤ p−1. Wenowhavethreepossibilities
for h .
2
SubcaseIII.2.1. Let h ∈/ Hp and h ∈/[H,α]. Wecanput h h ,insteadof h ,as
2 2 1 2 2
ageneratorof H . Inthiswayweobtainagroupthatis G-isomorphicto H . Thus
2 2
wegetthat[β,α]isinthisnewcopyof H .
2
SubcaseIII.2.2. Leth ∈/ Hp andh ∈[H,α],thatis,thereexistsγ ∈/ Hp suchthat
2 2
[γ,α]=h2. Putβ(cid:48)=βγ−1. Then[β(cid:48),α]=h1=α1p1a1−1d1. Hencethecommutator
chainofα iscontainedinthecommutatorchain(cid:104)(cid:104)β(cid:48)(cid:105)(cid:105)whichisanormalsubgroup
11
of G oftype3.
SubcaseIII.2.3. Leth2∈Hp; thatis,h2=(cid:81)i∈Bαip1ai−1di,where B={i:αi1∈H2},