On the Cayley Hamilton property in abelian groups

(1)

VOL. 15 NO. 4 (1992) 727-732

ON

THE CAYLEY-HAMILTON PROPERTY IN

ABELIAN

GROUPS

ROBERT R. MILITELLO

Rhodes College,

Department

ofMathematicsand

Coml)uter

Science 2000North Parkway

Memphis,TN 38112 Email: Militello(Rhodes (Received September 25, 1991)

ABSTRACT.

Inthis paper, the work ofCasacuberta andHilton ontheclassofabelianfg-like groups is extended. These groups share muchin common with the class offinitely generated abelian groups.

KEYWORDS. Localization, fg-like group, fgp group.

1991

AMS SUBJECT CLASSIFICATION CODES.

20K40, 20K15.

1.

INTRODUCTION

A

nilpotent groupGis said tobe finitely generated ateveryprime

(fgp)

if for each primep

thereisafinitelygeneratednilpotent group

M

such that

Gp

_

Mp.

If thereexistsasingle finitely generated group

M

which works for allprimes, then wesay Gis fg-like, or more specifically,

M-like. Here

Gp

denotes thep-localizationofG.

In

this paper,wecontinuethestudyinitiated

byCasacubertaandHiltononproperties of fypgroups.

In

particular,wefocusourattentionon annihilator properties,inthesenseof the Cayley-Hamiltontheorem,whichan abeliangroupmay

satisfy.

Cacuberta

andHilton

[2]

showed that if

A

isaspecialfgpgroup, the followingareequivalent:

(a) A

isfg-like;

(b)

V:

A

A,

3 non-zero

F

Z[t]

with

F()(A)

0;

(c)

V:

A

A,

:t monic

F

[t]

with

F()(A)

0;

(d)

V:

A

_

A,

3non-zero

F

(5

Z[t]

with

F()(A)

0;

(e)

’:

A

A,

3demonic

F

[t]

with

F()(A)

O.

Here,

anilpotent groupGiscalled special ifTG G FG splitsontheright, whereTGisthe

torsionsubgroupandFGisthetorsion-freequotient. Foranabelian group

A,

this isequivalent

to

A

_

TA

(R)FA. Weshould also mentionthat ademonic polynomial is a polynomial whose leading coefficient and constant coefficientare +1.

Thetheoremisproven byshowing the following implications:

(c)

()

(b)

=

(el) (o)

(2)

728 R.R. MILITELLO

there exist fgp groups whicharenon-special

(and

hence arenot fg-like) which satisfy property

(d).

Casacuberta and Hiltonhave shown the existenceof suchagroup.

In

thispaper, weshow that there doesexist a largeclass offgP groupswhichare non-special andwhich donot satisfy property(d).

We will say that anabelian group A is Cayley-Hamilton if it satisfiesproperty

(c).

We will

call

A

almost Cayley-Harniltonif it satisfies property (d). Obviously, Cayley-Hamilton implies almost Cayley-Hamilton; however, it is easy to see that the converse is not true. Thus, let

A

@

Z/2.

Then

F(t)

2t annihilates forasly

"

A

A.

However,

ifwelet the th _copy

of

Z/2

be generatedby x,, then

"

A

A,

given by (x,) x,+, cannotbeannihilatedbyany

polynomialwhich isnot amultiple of2.

In

Section 2, our goal is to show that if

A

is an abelian group whose torsion-free quotient

is Z-like and if, for aninfinite number ofprimes p,

TA

contains either

Z/p

"0’)₍₉

Z/p

"() _or

Z/p

"()_@

Z/p

’()+1 _{as a}_direct

summand,

_then

A

_is_not _{almost Cayley-Hamilton.}

In

_Section_3, wegeneralize the results of Section 2 byreplacingthe condition that the torsion-free quotientis

Z-likeby theconditionthat thetorsion-freequotientis

Z’-like.

Ofcourseit isnaturalto conjecture

that the theorem ofCasacubertaand Hiltonremains true ifoneconsiders anyfgpgroupswhich are specialorhave,foraninfinitenumber of primes, non-cyclictorsioncomponents. One should notethatourresults donotrequire that

A

is

f9P.

We onlyassumethat thetorsion-freequotient

is

Z-like.

We call such groups almost fgPand wedescribe someproperties of thesegroupsin

Section 3.

In

aforthcomingpaper,theauthorwill discussageneralizationof the Cayley-Hamilton prop-ertyinthe classof solvable groups.

This paper is basedonresearch thatformed part of theauthor’s dissertation at SUNY

Bing-hamton. Special thankstoProfessorPeterHiltonfor advising thedissertationresearch.

2. ON ALMOST fgp

ABELIAN GROUPS

OF

RANK

ONE.

Let

A

beanabelian group. We denoteby

[A]

the class of theextension

TA

A

FA

in the abeliangroup

Ext(FA,

TA).

Wewill sometimeswriteTforTAand

F

forFA.

Initially, wewillplacenorestrictionson

T

TA. However,

wewillassumethroughoutthat

F

FA

is

Z-like

for somepositive integer k.

We

will refer to such abelian groups asbeing almostfgP.

It

isclearthat analmostfgP groupisfgpif andonly if

T

TA

isfinitefor each primep.

Ourfirst objectiveistocompute

Ext(F,

T)

when k 1. Todothis,weidentify

F

withagroup

(1__

of pseudo-integers

(see

[5]).

Explicitly, let

F

p0,/I

all primesI),where

k(p)

isanon-negative integer for each prime

p)

C_

Q.

LEMMA

1. IfFisas above andTistorsion,then Ext(F,

T)

T/()T

PROOF.

Consider theexact sequence

Z

F

F/Z

Z/p

This sequence inducesthefollowingright-exactsequence:

Hom(Z,T)L

Ext(@Z/pk(P),T)

Ext(F,T).

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However, Ext(,Z/p

k(’),

T)

II

Ext(Z/p

k0’),

T)

IIT/p’(’)T

HT/p*()T

Further,Horn(Z,T)

P p P P

T

T,

dA:

T

HT,/p(.),

isthe canonicalmap.

Hence a(T)

T/p()Tp.

Since

p p p P

the sequenceisright-exact,it follows thatExt(F,

T)

Q.E.D.

Obviously,if

T

isalmosttorsion-free,then

Ext(F,

T)

0.

Hence, A

T

(

F

and

A

isspecial. Ofcoursethisresult followsfromatheoremofCasacuberta andHilton

[1]

which states that if

A

isB-likeand

T

isalmosttorsion-free, thenExt(A, T)

_

Ext(B,

T).

We formallystate this result as:

THEOREM2.

An

almost fgpabeliangroupofrankonewhichisalmost torsion-freeisspecial.

An

application of

Lmma

1.5inCasacubertaand Hilton

[2],

givesthefollowingcorollary:

COROLLARY

3.

An

almost fgpabelian group of rank one which is almost torsion-freeis

,Cayley-Hamilton

(almost

Cayley-Hamilton)if andonlyif

Tp

is Cayley-Hamilton

(almost

Cayley-Hamilton)

for each primep.

We now explore property

(d)

in the class of almost fgp groups of rank one which are not

almosttorsion-free. Wewillwrite

[(p)]

for

[A]

where_wpE

T,

and isthecosetof

w

modulo

pO’)Tp.

The next resultserves as apreludeforthemain theorem ofthis section.

THEOREM

4.

Let A

be anabelian group and let

F

be Z-like.

Suppose

that, for infinitely

manyprimes p,

Tp

-7(:

0and

p

0. Then

A

isnot almost Cayley-Hamilton.

PROOF. We

adaptanargumentofCasacubertaand Hilton in

[2].

Let

pl,p2,.., bean

enu-merationof the primeswiththe statedproperty.

Let

)q,2,... beasequence ofpositive integers

1 ifp, A, suchthat ehpositive integer occursinfinitelyoften.

As

in

[2],

set

,

A,

if p,

A,

and define

,"

T,

by

,

(z)

,z.

Ifqisaprime not included in

te

aboveenumeration,let

q

id:

Tq

-

Tq.

Let

’=

"

T

T. Notethat

’

isanautomorphism ofT. Since 0

fop ,p=,... =d id fo

#

,,...

i foow

m

’.[]

[].

Wh

’

d

toanendomorphism of

A

which inducesthe identityonF. Thus isanautomorphism ofA. Nowlet Gbeanon-zeropolynomialover

Z

with

G()(A)

0. Then

G(’)(T)

0. Let n be

apositiveinteger. Note that n

,

for infinitelymany i, sayj,j,.... For anygivenj inthe previoussequence, let _xj E

Tp,

-0. Then0

G(’)(xi)

G(p,)(x,).

Since

x,

#

0, itfollows

thatp

G(p).

Hencefor j jl,j2,.., itfollows that_pj

G(n).

Hence

G(n)

0. It follows

that G 0and

A

isnotalmostCayley-Hamilton.

Q.E.D.

COROLLARY

5. Let

A

be an_{abelian group and let}

F

_{be Z-like.}

Suppose

that, forinfinitely many primesp,

Tp

0 whilek(p) 0. Then

A

is not_{almost Cayley-Hamilton.}

(4)

730 R.R. MILITELLO

Wenowplaceconditionson

T

inordertoproveourmainresult anditscorollary.

THEOREM6.

Suppose

that for infinitely many p,

T

isnot cyclic, and k(p)

#

O. Call this setof primesS.

For

eachp(

S,

write

Tg

Z/p

’n0’)

Z/p’()2

_(finite_abelian

p-group),

_where, for each p, ,,,(p) and n(p) are positive integers with n(p)

>_

re(p). Let

(x,)

Z/p’"(p) _and

(y)

Z/p"(’)_and_write

w,

(rxp,

qpy,...)

forsomeintegers rpandqpwhichdependonp. If

anyoneofthefollowingconditions hold,then

A

isnot almostCayley-Hamilton. (a) qpor

r

may bechosentoequal0 forinfinitelymaay p S.

(b) qp, rp may bechosensothat

[%[,

_< [rp[p

for infinitelymany p S.

(c)

qp,

r

nay be chosensothat

[r]

+

n(p) <_

]qp]p

+

re(p)forinfinitelymanyp S.

Here

[tip

is theexactpowerofp which dividest.

PROOF. Ifeither qp 0 orrp 0 for infinitely many p

S,

then a generalization of the argumentinTheorem4shows that

A

is notalmost Cayley-Hamilton.

Suppose

that

(b)

holds. Let

S’

C_ Sbe theset of primes mentioned in

(b).

Let P,P2,...,be anenumerationof the primesin

S’.

Let .k, 2,... bea sequence of positive integers such that

1 if p,

lA,

For

each in

ea’h

positiveintegerappears infinitelyoften.

Let

tt,,

Ai

if p,

A,

primeP

S’,

define

Z/p

"{)

Z/p

m{’)_so_that

%(y)

r(1

pp)x.

_Condition

(b)

_{and the fact}

definedbythematrix

[MP0

I0]

Again,it iseasytoseethat isanautomorphism of

T.

For

p

’

S’,

let =id. Let

’

E"

T

T. Weobserve that

’

is an automorphism ofT. The formula

%(y,)

rp(1

#p)Xp

guarantees

w,

Wp. Hence

.[A]

[A].

Thus

’

extends to anautomorphism

"

A

which induces theidentityonF.

Finally, notethat

(x)

ttxp.

As

in theproof of Theorem4, theredoesnot exist a non-zeropolynomialover

Z

which annihilates

’.

Hencethere doesnot exist suchapolynomialwhich annihila.tes

.

Hence

A

is not almost Cayley-Hamilton. The proofusing condition

(c)

issimilar and dependson defininga homomorphism

,

Z/p

"()

Z/p

’*0’) _with _properties _{similar to} those of above.

COROLLARY

7. Supposethat, for infinitelymany p,

T

containseitherZip’’)_q)

Z/p

’0’) _or

Z/p’*()

Z/p

"()+ asadirect summand. Then

A

isnot almostCayley-Hamilton.

Nowlet

Tp

Z/p

(wpl

foreach primep. Let

F

<[

all primes

p)

azadlet .4 be theabelim,

groupdefined by

[A]

[(wp)]

e

Ext

(F, T).

Itwasshown by Casacuberta andHilton

[2]

that

A

isCayley-Hamilton. The following lemma shows that

A

(R)

A

satisfies thehypothesisofCorollary 7. Thus,the tensor

product

ofCayley-Hamilton groups need notbealmost

Cayley-Haznilton.

LEMMA

8. Let

A

and

B

be almostfgp groupswhere

FA

is

Z-like

and

FB

is

iEt-like.

Then A(R)Bisalmost

f

gp where

F(A(R)B)

is

lE’-like.

Moreover,

T(A(R)B)p

TATB(R)(TA(R)TBp).

PROOF. Note

that

A

Z

@

TA

md

Bp

Z

(R)

TB.

Then(A

;:)B),

A,

(R)

B

(R)

TAp)

(R)

(Z

@

TBp)

(Z

(R)

Z,)@

(TA

(R)

Zp)(R)

(Z

(R)

TB)

(TAp

TBp).

(Z

’

TA

6)

TB

(TA

(R)

TBp).

(5)

The result followssince

T(A

(R)

B)p

T((A

(R)

B)p)

andF(

A

(R)

B)

F((A

(R)

B)).

Q.E.D.

An

amusingconsequence of

Lemma

8isthat if

A

and

B

arealmostfgp andneither

A

or

B

is

torsion,

then,

if

A

(R)

B

is fg-like,itfollows that

A

and

B

arefg-like. 3. ON ALMOST fgp

ABELIAN GROUPS.

We now seek to generalize the results of Section 2. Given a

Z-like

group

F,

we choose a reducedrepresentation

R.

for

F

in thesenseofCasacuberta andHilton

[1].

Thenwemayreplace

F

bythesubgroupof

Q’

generated bythe column vectors of

R

for each primep. Casacuberta and Hilton have shown

[1]

that

F

contains Z

.

The quotient

F/Z

is a torsion group.

Moreover,

F

isgenerated sbythe cosets containing thecolumnvectors of

Rp.

EXAMPLE

9.. Let

R=

[1/p

1

0 lip Then

r

((1,

)

+

Z,(-,O)

+

Z

)

Z/p@

Zip.

EXAMPLE

10. Let

R

lip₀ lip₁ Then

r--

((

1)

+

Z

2)

Zip

.

In

general, wemay write

ff,

_

Z/p

’’(p)₍₉

Z/p

’’()_{(9"" (9}

ZIp

n() _where

n(p)

<_

n2(p)

_{_(}

<_ n(p)

andsomeof the factorsmaybetrivial.

LEMMA

11. If

T

is torsionand

F

is

Z

-like,

then

Ext(F,T)

@[T,/p",(P)T,

@...(9

PROOF.. Considertheexact sequence

Z

’

F

@.

This sequence inducesthefollowing right-exactsequence:

However,

’

Ext(@,

T)=

1-IExt(F,

T)=

IIExt(Z/p

",()

Z/p"’(P),T),

p p

H[Ext(Z/p",(),

T)

(9"-O

Ext(Z/p

"(),

T)].

H[T/p"’()T

,...

T/p""()T]

Also

Hom(Zt,

T)

T

(@T)

and A

.(@Tp)

I-I[T/p"X(p)T

@...

@Tp/p"()T]

isthe

p p p

canonical map.

Thus

ZX((OT,)

)

,[T,/p

"l(’r,

,...,

T/p’O’T,].

_{Since the above sequence}_is_right_exact,

theresult follows

.E..

Onceagain,itisclear fromLemma11 thatifTisalmost torsion-free, then .4isspecial. Hence

(6)

732 R.R. ILITELLO

THEOREM12..

An

almostfgp,almosttorsion-freegroupisspecial.

COROLLARY

13..

An

almost fgp, almost torsion-free group is Cayley-Hamilton

(almost

Cayley-Hamilton)ifandonlyif

T,

isCayley-Hamilton

(alnost

Cayley-Hamilton)foreachprime p.

We

also obtain generalizations ofTheorem4 andCorollary5.

THEOREM 14..

Suppose

that,for infinitely many primesp,

w,

0while

Tp

0. Then

A

is

notalmost Cayley-Hamilton.

COROLLARY

15..

Suppose

that, for infinitely many primes p,

n(p)

>

logp

exp(Tp)

and

T,

:f-

0. Then

A

is not almostCayley-Hamilton.

Now

weseek to extendCorollary7. Before doingthisweneedtoplaceanadditionalhypothesis

on

.

DEFINITION 16..

Let

R, be a reduced representationfor F. R, iscalled collapsible iffor each primep,

,

iscyclic.

In

thiscase,

_

Z/p

"(). If

F

is anarbitrary

Z-like

group, then

wecall

F

collapsible ifthereexistsareduced representation of

F

whichiscollapsible.

Note that all Z-likegroups are collapsible. As a result, the following theorem generalizes Corollary7.

THEOREM

17.

Suppose

that

F

is

Z-like

fork

>

1and that

F

iscollapsible.

Suppose

that, foraninfinitenumberof primes p,

T

containseither

Z/p"()

@

Z/p"(P)

or

Z/p"()

@

Z/pn()+

as a directsummand. Then

A

isnotalmostCayley-Hamilton.

Weclosebymentioning that the class of almostfgpabelian groupsisasomewhatwellbehaved class ofabeliangroups. Itisstraightforwardtoshowthat the classisclosed under theformation

of quotients, subgroups,tensorproducts,torsionproducts,andhomology.

However,

in contrast

to the classof fgp abelian groups, thisclass isnot a Serre class becauseit fails to satisfy the

conditionof closure underextensions. For example, intheexact sequence

Z

Q

Q/Z

it is

clear that

Z

and

Q/Z

arealmost fgp; however,

Q

isclearlynot almostfgp. We arepresently exploringpropertiesof nilpotent groups whosetorsion-freequotientisfg-like.

REFERENCES

1.

CASACUBERTA,

J. and

HILTON,

P.,

On theextended genus of finitelygenerated abelian

groups, Bull. Soc. Math. Belg., Sr

A,

41, n.1

(1989),

51-72.

2.

CASACUBERTA, J.

and

HILTON,

P.,

Onspecial nilpotent groups, Bull. Soc. Math. Belg.,

Sr B,

41, n.3

(1989),

261-273.

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CASACUBERTA,

J. and

HILTON,

P. Onnilpotent groups whicharefinitelygeneratedat everyprime,

(to

appearinExpositiones

Mathematicae).

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HILTON,

P.,

On

all sorts ofautomorphisms

A.mer.

Math. Monthly9,

(1985),

650-656. 5.

HILTON,

P.,

On

groups ofpseudo-integers,

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Math. Sinica

(new

series)

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(1988),

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HILTON,

P.,

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class of nilpotent groups,

(to

appearin the Journal

of

Pure

and Applied

Algebra).