On almost finitely generated nilpotent groups

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 3 (1996) 539-544

539

ON

ALMOST FINITELY

GENERATED

NILPOTENT GROUPS

PETERHILTON

DepartmentofMathematicalSciences StateUniversity ofNewYork

Binghamton, NY13902-6000USA

and

Department

ofMathematics

University of CentralFlorida

Orlando, FL32816-6990USA

and

ROBERTMIUTELLO

Department

of Mathematics RhodesCollege Memphis,TN38112-1690USA

(ReceivedOctober3,1994)

ABSTRACT. Anilpotentgroup Gisfgpif

G,

isfinitelygenerated (fg)as ap-localgroupfor all primes

p; it isfg-likeifthereexists anilpotentfg group HsuchthatG_v H_vfor all primesp. Thefgpnilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian

groups,asubgroupofanfg-likegroupisfg-like, andanextensionofanfg-like groupbyanfg-likegroup

isfg-like These properties persist fornilpotent groups with finite commutator subgroup, but fail in

general.

1991AMSSUBJECT CLASSIFICATION CODES: 20F18, 20K24,55P15

0. INTRODUCTION

Inearlierpapers the authors havestudied variousaspects of thetheoryof almost finitelygenerated nilpotentgroups

(see,

eg,

[CH

1, 2,

3;HM]).

If

A

isan abeliangroup,wesaythat

A

isfinitely generated

ateveryprime(fgp)if, for allprimes p,

A,

isafinitelygenerated (fg)

Zr,-module.

Wealsosaythat

A

is

fg-like ifthere is anfgabeliangroup

B

such that

Ap

=

B,,

for all p. We also saythat

A

is B-like Obviously an fg-like abeliangroup is fgp, but the example Zip shows that the converse isfalse. In

[CH]

theauthors effectivelycharacterizethe fg-likeabeliangroups amongthefgpabeliangroups

A

for

whichthetorsionsubgroup

TA

is a directsummand;andthe storyistaken furtherin[M1 ].

Itisstraightforwardtogeneralize thenotionsfgp,fg-liketonilpotentgroups. The generalization of fg-like is immediate; astothe generalization offgp, we need the concept ofa setof generators ofa

p-local (nilpotent) group. Thus theset S

c_

H,

where

H

is ap-localgroup generates

H

as p-local groupifHisthe smallest p-localsubgroupofHcontaining,.q. Equivalently, let

(S)

be thesubgroupof

H

generated by S. ThenSgenerates

H

asp-local groupif

(S),

H,

andwesaythatGisfgpif, for all

p,

G,

isfinitelygeneratedasp-localgroup

Nowitis not difficulttoshow thattheclass offgpnilpotent groupsconstitutes aSerreclass in the senseof[HR] Thatis,the abelianfgpgroups formaSerreclass inthe usual sense, but thebasicaxiom isgeneralizedtoassertthat,for any shortexactsequenceofnilpotentgroups

G’

G

--

G",

(0.1)

540

Inthispaperwe areprincipally concernedwith this basic axiom and itsanalogue for fg-like groups Theother axioms create noproblemsin eithercase,since tensorproduct,torsionproductandhomology (in positivedimensions)commutewith localization

In fact, the basic axiom fails in one particular case, even for abelian groups. For let

A

be the subgroup ofQgenerated bythe rationals _?,all p Then

A

isZ-like, buttheembedding

Z

c_

A

induces a

shortexactsequence

Z

A

--

(Z/p, (02)

p

where the quotient, asalready mentioned, isnot fg-like

However,

for abeliangroups, the othertwo assertionsof thebasic axiomfor fg-like groups do hold We may putthis inthe following way

THEOREM0.1. Inthe category ofabeliangroups

asubgroup ofanfg-likegroupisfg-like,

ii anextensionofanfg-likegroup byanfg-likegroupisfg-like

Wethus devote muchattentiontothestatusof theextensionof Theorem01tonilpotentgroups Theplanof thepaperisasfollows InSection weprovethat the class offgpnilpotentgroupsis a

Serreclass,mostofthishas appeared alreadyinthe literature(see[H2]),butwehave feltitdesirableto make thispaperlargelyself-contained InSection 2 wegive the(easy) proofof Theorem01 InSection 3 we show that Theorem01 does extendto an interesting subclass ofthe class of nilpotent groups, namely, tothe class of nilpotent groupsGwith finite commutatorsubgroup

[G,

G]

Itisinterestingto

remarkthat,forfgpnilpotentgroups

G,

thisclass oinides withthe class of thoseGsuch that

G/ZG

is

finite, whereZGisthe centerofG.

In

Section 4 weshow that both parts of theanalogueofTheorem0.1 failfor the classof nilpotent groups Indeed,we construct atorsionfree nilpotent fg-like group

H

ofrank2,admittingasubgroup

G,

such that (i) Gis notfg-like and(ii) Gis an extensionofanfg-likeabeliangroup byanfg-likeabelian

group Althoughourconstructionisspecific,wepointoutthat the construction methodcanbe applied

toproduceacontinuum ofsuch counterexamples.

Weclosethisintroductionbyexplaining the genesis of theideasoffgpand fg-like nilpotent groups

inhomotopytheory Mislin

(see [M2])

introducedthe idea ofthe genus

of

anfgnilpotentgroup Nand thatof thegenusof a nilpotent spaceof finite typeX. Thus thegenus of

N

isthesetof isomorphism classes offgnilpotentgroups

M

suchthat

N,

M,

for allprimesp; and thegenusof

X

is thesetof homotopytypes ofnilpotentspaces

Y

of finite type such that

X,

Y,

forall primesp. Wesay that

M

is in the genus of/g and that

Y

is in the genus of

X,

writing ME

(N),

Y

(X)

Plainly if

Y

E

(X),

then

r

Y

E

(r

X),

where

r

isthefundamentalgroup.

Thefiniteness restrictions onNand

X

aredesignedtorender

(N)

and

(X)

calculable- indeed,

(N)

willbefinite, and

(X)

willbefinite if

X

iscompact. However,thereis no reason inprinciple for these finiteness restrictions. In particular, the finitenessrestriction onN rendersthe notion ofgenus uninteresting in the case that N is abelian, since then

(N)

is trivial.

In [H1]

the author initiated a

"controlled departure" fromthisrestriction,which was carriedfurtherin[CH1],westillrequireNtobe fgbutallow

M

tobe arbitrary. Thisledtothe notion of the extendedgenus; and,of course, the extended genus ofNisjust thesetofisomorphismclassesofN-like(nilpotent)groups,and the questionarisesof howtocharacterizefg-like nilpotent

(or

abelian) groupsinthe classof allnilpotent

(or

abelian) groups.

Sincethe notion offg-lke is so closely relatedto localization

(at

every prime), itwasnatural to

introduce, aswehave doneinthispaper, theideaofgroupswhicharefgatevery

prone

intothe attempt

tocharacterizefg-like groups Whenitturnedoutthatthisclass ofgroupssatisfied theSerreaxioms,we

becameconvincedthatit wasreallyworthstudying.

ONALMOSTFINITELY GENERATED NILPOTENTGROUPS 5/11 typewhich was a circlebundle Initsextended genusonefound nilpotent spaceswhich were

K(Tr,

1)-bundleswhere7ris agroupofpseudo-integers [HI ], thatis, agroup intheextended genus ofacyclic

infinite group Ofcourse, justas forthegenus, ifYis in the extendedgenus ofX,then7rYis inthe

extended genus of7rX

Notice that the notionsfg-like andfgparerelatedby "commuting quantifiers ThusNisfg-likeif

qM.Vp.Mp

-

Np,

andNisfgp

ifVp.3M.Mp

-

Np

(Here Mis, of course,fgnilpotent

1. THE SERRE CLASSOFFGPNILPOTENTGROUPS

Herewereviewtheargumentswhichshow that the class offgpnilpotent groups formsaSerreclass

inthe extended senseof[HR],recall thateffectivelythisjustmeansthat theabelian fgp groupsforma

Serreclass intheusual sense andthat, giventheshortexactsequence

G’HGG"

(1 1)

of nilpotent groups, thenGisfgpifandonlyif

G’

and

G"

arefgp

Aspointedoutinthe Introduction,anilpotent group Gisfgpifandonlyif, for each primep,there

exists anfgnilpotent groupH such that

Hp

Gp,

indeed,we mayevenassume that

H

c_

Gp

so that

H,

G,

This makes it obviousthat the class ofabelianfgpgroups is closed undertensorproduct,

torsionproduct and homology, sincethese constructions commute with localization and preservefinite

generation Indeed,for thesamereason,theclass of nilpotentfgp groupsisclosed underhomology. We alsoclaimthat thebasicaxiomofaSerre class,relatingto

(1 1),

obviouslyholds forabelianfgp groups.

Wenowprovethebasic axiom ingeneral

THEOREM1.1. Let

G’

G--,

G"

beashortexactsequence of nilpotent groups. ThenGisfgp

ifandonlyif

G’

and

G"

arefgp.

Webaseourproofon twoimportant lemmas

LEMMA 1.2. Let N G

k_.,

Qbeacentralextension withGnilpotent. ThenGisfgpifNand Qarefgp

PROOF. Letp be a fixed but arbitrary prime and let

H,

L befgsubgroups of

N, Q

such that

H,

N,,

L

Q.

Of course,

N,

Gp

--

Q

iscentral. LetMbeanfg subgroupof

G

mapping

ontoLand letK

(H,

M).

Weclaimthat

K,

Toseethis, letx E

G,.

Then thereexists

a/f-number

q such that

kpX

q_E

L

_sothat

kx

ky,

for

some y

M,

andxq _{yz, with}z

Np

Thenthereexists

a/f-number

r suchthat z

H,

whence

xqr _(yz)

yz

_{K. Thisshows thatx E}

K,

_{and completestheproofofthelemma}

REMARK. Of course, theconverseofLemma 1.2 alsoholds- indeed, itfollows from Theorem

1.

However,

wedonotneed theconverse toproveTheorem1.1.

Oursecond lemmainfact exploits Lemma1.2

LEMMA1.3. LetGbenilpotent. ThenGisfgpifandonlyif

G

isfgp.

PROOF. We alreadyknow that

G,

isfgpifGisfgp,since

Go

Hi(G). Suppose,

conversely, that

G,

isfgp. Weconsiderthe Hallcommutatormap

.

(R)’G

Now (R)’G,isfgp,so,bythe knownfacts for abeliangroups,

F’-I(G)/F’(G)

isfgp.

Consider nowthe centralextension

F’-(G)/F’(G)

G/F’(G)

--,

G/F’-’(G).

Wearguebyinduction on that

G/F(G)

isfgp. This is truebyhypothesisif 1, and theinductive

step from

(i

1)

to isjustanapplication ofLemma 2 to thecentralextensionabove. We completetheproofby taking sufficiently large thatF

(G)

is trivial.

542

Wereturnnowtotheproofof Theorem SupposethatGisfgp ThenG,bisfgp,so

G,

isfgp and, by Lemma 3,G" isfgp

Now let p be anarbitrary prime and let H be an fg subgroup of

Gr,

such that

Hr,

Gr,

Then

H

n

G,,

as asubgroup of anfg nilpotent group, isfg and

(H

n

G)

r,

Hr,

f3

G,

G,

Thisshows

that

G’

isfgp

Finally, suppose G’and

G"

bothfgp Wehavethe shortexactsequence ofabeliangroups

andasurjection

G’

--

G’

/G’

f’l

[G, a].

Since

G’b

isfgp,so is

G’/G’

[G, G].

Since

G’/G’

Ca

[G, G]

and

G’.’b

arefgp,so is

G,,

Wenowapply Lemma 3 tocompletetheproofthatGisfgp

2. THE CLASS OF FG-LIKE ABELIAN GROUPS

We knowthat the subclass of the class offgp nilpotentgroups, which consists of fg-likegroups, doesnotformaSerreclass. The difficulty doesnotlie withthesupplementaryaxioms,which continue to

be valid, forthe same reasons asbefore

However,

thebasic axiomfails,evenfor fg-likeabeliangroups, since,aspointedout intheIntroduction,ahomomorphic image ofanfg-likeabeliangroupmayfail tobe fg-like However,insofar as abeliangroupsareconcerned,this istheonlypartof thebasic axiom which fails This is aconsequence of the followingtheorem

THEOREM 2.1. Let

A

be anabelian group Then

A

isfg-likeifand onlyif

A

isfgpwithTA

finite.

PROOF. Let

A

be fg-like. Then

A

iscertainlyfgp

Moreover,

if

A

isB-like, with

B

fg, then

TA

"

TB,

which is a finitegroup

Suppose

conversely that

A

is fgpwith

TA

finite. Then

FA

is fgp, so that

FAr,,

for a fixedbut arbitrary primep, isthe p-localization ofafree abelian group

Z

’

offiniterank Moreover,

FAo

Qk,

so that k is independent of p. Thus

FA

is

Zk-like.

Now

TA

is finite. Thus, as in [CH1],

Ext(FA,

TA)

0and

A

TA

FA

Itfollows that

A

isB-like, whereB

TA

Z

k

REMARK. Thefirst, easierpartofthe argument, guaranteeing thatanfg-likegroupisfgpwith finite torsionsubgroup,plainly extendstonilpotentgroups

COROLLARY2.2. Let A

-

A

k.

A"

beashortexactsequenceofabeliangroups. Then

i.

A’

isfg-likeif

A

isfg-like;

ii

A

isfg-likeif

A’

and

A"

arefg-like.

PROOF. (i)Ofcourse

A’

isfgpif

A

iffgp.

Moreover, TA

is asubgroup of

TA

and henceisfinite if

TA

isfinite. Thus(i)follows from Theorem2

(ii) Ofcourse

A

is fgp if

A

and

A"

arefgp. Moreover

TA

TA

k(TA)

is a shortexact

sequenceand

k(TA)

c_

TA".

Thus,since

TA’

and

TA"

are finite,sois

TA,

and(ii)also followsfrom Theorem2.1.

The rest ofthis paperis primarily motivated byour seekingto answer the question whether the analogue of Corollary2.2holdsfor nilpotent groups. Wewill see in thenextsectionthatitdoesholdfor animportant class of nilpotentgroups,and inSection4thatitdoesnotholdingeneral

3. ON NILPOTENT GROUPS WITH FINITE COMMUTATOR SUBGROUP Wehaveprovedelsewherethefollowingfundamental theorem

[HM]

THEOREM3.1. Let N G

--

Q

beashort exactsequenceof nilpotentgroups,withNfinite Then

G

isfg-likeifandonlyif

Q

isfg-like.

ONALMOSTFINITELY GENERATED NILPOTENT GROUPS 543

COROLLARY:3.2. Let G bea nilpotent groupwithfinite commutatorsubgroup ThenGis

fg-likeifandonlyif

Gab

isfg-like

Wenow establishthe analoguesof thetwo parts ofCorollary2 2 for nilpotent groupswith finite commutatorsubgroup

TtlEOREM 3.:3. LetG’beasubgroupof anilpotent groupGsuchthat

[G, (7]

isfinite ThenG’is

fg-likeifGisfg-like

PROOF. SinceG,bisfg-like,sois itssubgroup

G’/G’

f3

[G, G]

But

G’

n [G, G]

isfinite,so

G’

is

fg-like by Theorem3

REMARK. Obviouslyit suffices to assume

Gab

fg-like and

G’

n [G, G]

finite

THEOREM3.4. Let G’ G

G"

be ashortexact sequenceofnilpotent groupswith

[G, G]

finite Then if

G’, G"

arefg-like,so isG

PROOF. Considertheshortexactsequenceof abeliangroups

c’/c’

It,

c]

c

-

c’o’.

Since

C’

rq

[C, C]

isfinite, itfollows from Theorem3 that

G’/(7’

rq

[G,

C]

_isfg-like Also

G,

is fg-like,so,by Corollary2 2(ii),

Gab

isfg-like Thus,by Corollary32,Cisfg-like

Itisnotsurprising, inthelightof Theorems33 and34that theanalogue ofTheorem 2 holds for nilpotentgroupswith finite commutatorsubgroup Wefirstpresentaneasylemma

LEMMA3.5. Let Gbenilpotentwith

[(7,

G]

atorsiongroup Then

T(a)

Ta/[a, a].

PROOF. Inthe shortexactsequence

TG/[G,

G]

G/[G, G]

G/TG,

thesubgroupis torsion and thequotientistorsionfree Thus

TG/[G,

G]

T(G/[G, G])

THEOREM3.6. Let Gbe nilpotentwith

[G, G]

finite Then, ifGisfgpwithTGfinite,Gis

fg-like

PROOF. By Lemma3 5

T(G)

is finite. Thus,since

G

isfgp,itfollows from Theorem2 that

Go

isfg-like So,therefore, byCorollary3.2,isG.

Of course, Theorem3.6could be made thebasisforalternativeproofsof Theorems 3.3,3.4

4. ACOUNTEREXAMPLE

Inthissection we construct anexampleofanfgpnilpotentgroup Gwhich is notfg-like butwhich

may be obtainedas anextensionofanfg-like group byanfg-like group WethenembedGin anfg-like group H. Thuswehaveacounterexampletothe generalization of each part ofCorollary22tonilpotent groups. Thegroup H(andhencealsothegroup

G)

is torsionfree andnilpotentof class2 ThusGalso providesacounterexampletothegeneralizationofTheorem2 tonilpotentgroups. Ofcourse, if sucha

generalizationhadbeen valid,wecould have generalized Corollary2 2also

Let

A

c_

Qbe the group ofpseudo-integers [H1 generatedbythe rationals

,

as p varies over all

the primes(seetheIntroduction). LetC

()

beacyclicinfinitegroup actingon

A

Zbythe rule

.(,)

( +,,,),

(4 )

and let Gbe the semidirectproductfor this action. Since

A

Z

andCaretorsionflee, Gis torsionflee, andGisplainlyanextensionof anfg-likegroupbyanfg-like group Ifwe write

A

Z

multiplicatively, thenGhas the presentation(withT, ’representing’

)

G

(Tp,

,

lTpTq

TqT-p,7-pC CTp,

Tp

Tp,

T;

T,

C

-1 Ta,Vp, q), (42) where

544

Itisplain from(42)that

[G, G]

(7-)

and that7-, E ZG Thus

[G, G] c_

ZGsothatGisnilpotent of class2 Furtherweseefrom(4 2)that

G,,

G/(’r)

(p,

,

[Y

1,Vp}. (4 4)

Thus

G,

hasp-torsionfor allp, so

G,

isnotfg-like Neither,then,is

G,

sothatGisanexpleofan

fgpnilpotentgroup of class2 which is an extensionofanfg-likegroupbyanfg-like group,but which is notitselffg-like

WenowembedGinHby addingto(4 2)newgenerators

,,

forall p, d new relations

Plainly

H

may be regarded as asemidirectproductforan actionof

A

on

A

Z

distherefore also

torsionfree Wenowhave

[H,H]

(rp,Vp)

and

ZH

(rp, Vp),

soHisnilpotentof class2

We nowfix

arbitra

primep and considerthe

locization Hp

ofH. The

locization

of Thus,

agn

writing

AZ

is

A,Zp,

where

A,

is the cyclic

Zp-module

generated by

?.

multiplicatively,withrprepresenting ?,

where

()p

meansthatthetesinsidethe braces provideapresentation of

Hp

as

lal

p Let K be thegroupgiven by the presentation

K

(r,

,

lr

r,

r

r,

-1

to).

(4.7)

Itisthen planthatKis fgnilpotentgroup of class 2, and

eompson

of(4 6), (4 7)shows that

Kp

H,,

Vp. (4 8)

Thus

H

is K-like,sothat

H

is fg-liketorsion

ee

lpotent oupof class 2 thasubgroup

G

wch

is notfg-like. NoticethatKisjust the

ee

lpotentoupof class2 on 2generators,K

F

(,

)

Of course,ourconstructioncould be itated th

A

replaced by yothergroupof pseudo-integers

notisomoctoZ

If

A=(

,Vp),

then

A/Z

Z/p(’) _Butthe

oe

_{of(4 4)tellsus that} thetorsionsubgroupof

G

isprecisely

Z/p

("),

sothat, we

v

A,

we

v

the

isomosm

classofG. Sincethereis a continuumofoupsofpseudo-integers

[H1

],thereis a continuumofoups Gwchcbeconstructedin

ts

way hangthepropeiesaafibutabovetoourspeci choice of

G.

In piculeach suchoup Gmay be embedded inasuitableK-likelpotent oup

H,

where

K

isthe

ee

lpotent oup ofclass2 on 2generators

NCES

[CH1

Casacubea,Cles dPeter lton, Ontheeended genus of fitelygeneratedabelioups, Bu. Soe. Mat. Belg. (Series A),

I,

(1989),51-72.

[C]

Casacubea

Cles dPeter

Iton,

On speci lpotent oups, Bull Soc. Mat. Belg. (Series

B),

I3

(1989),

261-273.

[CH3]

Casacubea, Cles dPeter lton, On lpotent groupswch e fitelygenerated at

eve

prime,ExpositionesMathematicae,10(1992),385-402.

[H1] lton,

Peter,

On groupsof pseudo-integers,Aa.Math. Sica4,2(1988), 189-192.

]

lton,Peter, Ontheeendedgenus,ActaMath. Sica4,4(1988),372-382

[H3]

lton,

Peter,

Ona flyofSereclasses oflpotent oups, Joum. Pure d

App.

g.

89

(1993),

127-133.

]

lto

Peter d Robe litello, Someremarkson

most

fitelygenerated lpotent groups, PublicacionsMatemhtiques,36

(1992),

655-662.

[]

lton, Peter d Joseph Roitberg, nerized

C-theo

dtorsion phenomena inlpotent spaces,Houston J.ofMath. 2(1976),525-559.

[M1]

lkello, Robe, On Cayley-Hlton Propeies in Cen Classes of

oups,

SY

Binton

(1991).