Sci.Rep.Kan品zaw
aUniv.~
Vo!
.22，No.1，pp. 9‑14June1977

theSchur:MultiplicatorsandCentmISubgr:oups
Kunihiko

0/Scie幻ce，J(anazawaUniversity(R日ceiv
edApril26，1977)

De
仰 げment

Abstr
ad Thεaimo
Ith
色 pr
esen
tpap
位 協 t
ogiv
esomeinequaliti号sforth日 ord位。
ftheS
chllrmu!
tiplicatorM(G) H'(G，T)ofaf
initgrollpG，wherεT=Q/，ι日ニ1. Iutroduct!ouTheJollow
ingnotationw
illbeused: eisthei
dentityelemen
tofG. IfA and Yaresubsetsof G，denotesasubgrou
pofG generated by theelementsCx，11yJ=X‑y‑xy，whereX EX andy臣 Y. Thelowercentralserie8 lG，f， i=O，1，2，"'， is= G，G1ニG'ニ ， GJ， Gi H= i，Gl Z(G)，or i8thecenterofG¥clefinedThe uppercentralseries 1Zd ，i=0，1，2，…
， is d巴fined{eI，Zj=iff
Gisnilpotentofclassn iffGn‑1ヰlef，01Ziは /Zi=Zn‑lヰ Zn=G. Tensorproduct③ istake
noverZ.
1owingi
nequalit
yforacentral8ubgroupilof usingthe
1nC
5JJonesgavethefol
freepresentation:
ァ勾A iiG' 1 •1M(G)1豆; 1M(A)1 ' 1M(G/A)11⑩(G/G'A11.

Ashestatedthere，thesamer
esultcanbeobtainedviathespectralsequence.Weshallprov
ethefollowing theorem as an improvement ofthe above，usin
変 exactseq，uences Iwahori‑Matsumoto(3JandinProposition2belovv.
THEOREM 1.

Forα centralsubgro坤 A 0/ ωehave

ハG'1 ・1M(G)1豆: 1 M(A/AiiG')1 ・1M(G/A)1 •IA(G/G11.

Onecanseethatmoreoveri
fG=G'andifG hasa'non‑triv
ialcentralsubgrou
pA，then wεhave 1A 1 ・iM(G)1 = 1M(G/A)1，inpar
ticular，G/A hasnon‑triv
iaimultiplicator.
8Jusedhi
sexactsequenc
eto have
Foranilpotεntgrou
pG ofclassn，VermaniC
自KunihikoSATO

10thefollowinginequality(SeealsoG
aschutz‑Neubuser‑Yεn[2J):

IG'I• I
M(G)I孟. I
M(G/G')I• I
G'Id(G/Z)，(Nj)

whered(G)istheminimumnumbero
fg
ener
ator
sofG.
Anotherestimategiv
enbyJones[5JisI
G'I• I
M(G)I
手. I
M(G/G')I• 1
1I(G/G') ⑧(GJG~) 1.

(N2)

Inthepresentpaper we shallfurthergiv
ethefollowin
gtwo inequalitie
softhisbtaine
dfromanexactsequenceofVermaniandt
heotherfromTheorem1.
natu
re，oneo
THEOREM2.

sGi(N3)I
G' I• I
M(G)I
豆 ￨M(G/CF)￨@ntll(C/GFZz)⑧ (GJGi+ 1.

ofclassn，thenwehave1)

THEOREM3. InthesameasSUn11
うがonasweh
αver︐fjz↓GZ8:ZzμrwhMZGM川トHF︐︐︐︐︐︐znGM/﹀
一GMGN2. ExadS母quencesbeliangroup，writte
nmultiplicati司rely，takena
saZ‑modulet
hroughLetA beana
r.a=αγforaE A andrEZ.Let^(A)denotethesecondantisymmetricp
roductofA，.
ie.thefactorgroupo
fA⑧ A byN(A)，where N(A) isgeneratedbyt
heelementsof'g)a andα(⑧ b)• (br'g)a). Wedenotebya八 bthecosetcontainingα⑧ b.
formarWhenA isanabeli
ansubgroupofagroupG，wes
halldefin
eahomomorphism¥
V
CAofM(G)toHom，)
asfollows:f anddefineα1:AxA TFirstt
akea2‑cocyclefo
予(a，b)=f(a，b)‑f(b，α)fora，bEA.

Thenα1is(1)andsincewehavee
asilyぱ)(a，bc)=f(a，，a)= ‑f(b，c)+f(ab，c)+f(α，b)‑f(c，a)‑f(b，ca)+f(b，c)a)=f(a，b)+f(ba，c)‑f(b， 一(a，c)‑f(b，a)一f(c，α)=f(a，b)+f=α( (a，b)+(af)(a，OntheSchurMultiplicator
sandCe持tr
alSubgroups0)'11(af)(bc，a)= 一(a/)(a，bc)=‑(α1)(a，b)一(ザ)(a，c)=(め (b，α)+(af)(c，a)
fora，b，cEA.
Thus afi
nducesず E Horn (A0A，T) suchthat(否rf)α(0b)ニ(口月 (a，b). Sinceafvani
shesonN(A)，iti
nducesず E H0111 C'(A)，T)sucht
hat(号i(a八 b)ニ(ぱ)(a，b)。
Finallyiffi
sacoboundaryofh，say，then(万月初 Ab)= (oh)(α，b)一 (oh)(b，σ)= (h(α)‑h(ab)+h(b))一 (h(b)‑h(ba}+h(α))=O.
andefine<;
IlC.
A(~)= ;
cfwherefisarepresentativ
ecocycleof;rE M(G).
NowwecPROPOSlTlO
N1. ForanabcliangroupA ，<;IlA.
Ai
sanisornorphisrnandwehave(2)^(A)，T).
M(A)三 Horn(PROゆF. SeeYamazaki[7，corollary2toTheorem2.
2].

EREMARK. Let an abelian group A be decomposed into昌 directproductAρ9where Ap isthe Sylowρsubgroup of A. Itiseasytosee M(A)三 ^(A)さ E1Aρ)
DenotebyZ(n) acycli
cgroupoforder n，andby jZ(n)fm thedirectp
roductofmrcopiesofZ(n). IfA = 2:Z(ρ勺 withCjミ
ミ C2ミ…主主 e，r then^(A)三 2: jZ(ρ 円 H H ，whichisawell‑knownr
esultofSchur[6].
Ther行str
ictionhomomorphismres→ M(A)forasubgroupA o
fG induces
A :M(G)C.
anexactsequenC(ぅ:(3)0→ M(G)A → M(G)→ M(A)whereM(G)A istheker
nelo
fresC，A' I
fA isabelian，thentherelation<;
IlC.
A=伊'A，A'res.
andProposition1implyker(<;IlC.
A)=ker(resC，A)'Hencewehaveanexactsequence:CA(3)'0→ M(G)A→ M(G)→ Hom((A)，T).
八PROPOSITION2. IfA isce
ηtralinG，theηωehaveane
xactsequence:(4)0→ M(G)A→ M(G)→ M(A/AnG').

PROOF. Insteadof(4)，weshallprov
ethefollowingexactsequence:(4)'0→ M(G)A→ M(G)→ Hom(^(AnG')，T).

Kuni
hikoSATO121)，weusedthefactsthat/i
8aonG andthatab= ba，Inthecomputation(ac=SinceaEACZ inthepresentcase，wehaveab= baforany bEG. ThereforetheextendedmaptoA xG ishencevanishesonA xG'. Thusa/vanisheson
A.x(AnG')andon(An >(A.，whichinduces:(A/An x /AnG')→T sllch(γ)α((Aハ ハ G'))ニα(刀 (a，b).
A byA/AnG¥andweh
aveathat。
JG，A ofM(G)toHom (吋A/AハT)quitesimilarlya
sweobtained伊G，A. Themapqexactnessof(4)'isobvious，since(1'/=0isequivalenttoαJ'=O.
Q.E.D.

COROLLARY. WehaveresC，
A =0if(i)AζZ
ハG'，.
ie. Gisasiemextension
A/AハG'iseveη
σA ，orREMARK. Thεcaseis Theoremof Vermani [9]，and thecase(i)i
810，Theoren
14.
tothecaseo
faweakstemextension Vermani[3. Px'i[JJi[)fofTheor倍ms¥Il1. Itfol
1owsfromPronosition2thatPROOFOFTHEOREl/Plハ G')I'I

三玉I.

exactsequeロceo
f1wahori‑M抗 sumoto

Moreoverthe，T)一→ Hom

O→ HomT)→ Hom(A，寸)→ M(G/A)→ ‑ )Hom((G/G'A)i2)A，T)

implies
三凡I(A)⑧ A I• I
M(G/Pl)I， 1 G' I

IA I• I)/{G/AYI￨(G/G'A)i2)A I IM(G/A)Iu

IAnG'I

Weobtainthεrεsultbγcombiningtheabovetwo
PROOFOFTHEOREM3. PutG =Sincewehavei‑1andA=Z(G/Zi‑1)=Z;/Zi‑linTheorem1.

ハ(Grzz1/Zzl)=ZZl⑩ni.
‑lriand)/(n主 Z;/(Zin
五三wehave

Q.E.D.

G'Z;/G'。
ηJ忌 Ii

Q/Z;̲，)I引 制 G/Zi) I'I
M(G'ZG'Z

即of:
1heaboveinequalitiesfori=1，2，・・ おThε13theSchur1
1
食dtzlうlicator
sandCentralSul習routs01pE.Dthe1己sul
L園

PROOFOFTHEOREM2. Tbeexactsequen
問 。fVermani[8J。
ーや HornT)→，T)→ → Hom ((G/ZimpliesIGn‑ I I
1I II
豆: I@1uI
fwe

G，; thenfZi‑'appropriateexpressionX/Gi oG;).
subgroup;ofwedonothavean
X of"¥i<ie canusethearesimilartothoseo
fTheor日mQ.E.D.

つJ.

札

4. Examp]昭島
we presenta，)， (N2)，
and(N(i) L
etG begene
ratedofto compare theupperhounds ofa，bandcwithd
efiningrelati倒 防 ;αTa戸β = (1)γ = [ム][b，c=ゲペC)e，whereρisaprimeandα，s，yandlJaresuchthat':1主 βミ y and0 >Oミa‑ y>O.
ThenG isofc
lass2，and(N，)， (N2)，andimplythebounds ，t5， andrespectivelyint
hecaseof仏( β，y，の1，1， andρ ヘρ ρ 5il1thecaseo
fα(，sJ ì/~6)=(2，2，1，LetG begeneratedα/l==e1[ai1・t.
‑'12，…，an withdefiningrelations:ニaj ρfori

wheren;;;2andJうisaprimeoGisofclass η ，and，ifρis odd，al1 thεfour
Jう=2，(N1nd 2)implythebound)aimplythe samevv
hile 3) and (¥14KunihikoSATO

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