A QUOTIENT OF THE SET [BG,BU(n)] FOR A FINITE GROUP G OF

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ijk ljklmno pn qm prsonqptuqo qvo kowuqpmrnvpl xoqyoor qvo vmzmqml{
|wunnon m} zuln }kmz }mk ur ~rpqo tkmjl qm ur qvo otkoo
|vuku|qokn m} qvo {wmy njxtkmjln m} _ vkmjtvmjq qvpn lulok yo ypww
woq xo u ~rpqo tkmjl u lkpzo pspprt qvo mkok m} ur u {wmy
njxtkmjl m} _ r upqpmr yo ypww jno vuk qm kolkonorq qvo noq
m} otkoo |mzlwo |vuku|qokn m} ur ypww woq vuk kolkonorq qvo
njxnoq |mrnpnqprt m} qvmno otkoo |mzlwo |vuku|qokn m} qvuq uko qvo
konqkp|qpmrn m} |wunn }jr|qpmrn mr _
o ypww ~knq on|kpxo u ruqjkuw zul
vuk
yvp|v ypww xo pn|jnno qvkmjtvmjq qvpn lulok _ oq jn ouzpro qvo }mwwmyprt
putkuz yvoko ol mz rr

F ¡ ¢F ¡ £
vuk ¤
ol vuk ¥

vo pnmzmklvpnz pr qvo jllok wo}q pn u konjwq m} ymk¦ x{ §u|¦myn¦p ¨|wjko
ur
iwpsok ^© yvpwo qvo pnmzmklvpnz mr qvo }uk kptvq pn u konjwq m} ymk¦ x{
ªy{ok ur «uxkmn¦{ ¬ ur qvo pnmzmklvpnz pr qvo wmyok kptvq pn u xunp|
konjwq pr kolkonorquqpmr qvomk{_ ®¯ pn qvo zul prj|o x{ qvo pr|wjnpmr m}
qvo {wmy lnjxtkmjln prqm _ o rmy woq zuln ° ur ±° xo qvo zuln
²
qvuq zu¦o qvo putkuz |mzzjqo _ o ypww nvmy pr qvo roq no|qpmr qvuq qvo
pzuto m} ° ur ±° xmqv wpo pr qvo njxnoq vuk vuk _

m yo ypww woq ur ± xo qvo zuln ° ur ±° konlo|qpsow{
ypqv qvo

kurto konqkp|qo qm vuk _ my qvuq yo vuso o~ro qvo zul

yo nquqo qvo }mwwmyprtqvkoo qvomkozn yvp|v olkonn qvo zupr konjwqn m} qvpn
lulok
! vuk ¥

" # $ % % ! vuk ¥

& vuk
$
'o}mko tmprt mr yo nvmjw rmqp|o qvuq }mk qvo putkuz uxmso qvo zuln
o|wjprt zuln }kmz |ur ou|v xo nolukuqo prqm u lkmj|q
msok uww %% % {powprt u nolukuqo putkuz }mk ou|v %% %

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¤
vuk ol ¥
o ypww ymk¦ ypqv ou|v lkpzo pspprt qvo mkok m} nolukuqow{_ )pknq yo
ypww nvmy qvuq qvo pzuto m} qvo zul ±°* pn uwyu{n |mrqupro pr qvo njxnoq
vuk vuk _ ir|o yo vuso mro qvpn yo ypww woq ±* xo qvo
zul ±°* ypqv qvo kurto konqkp|qo qm vuk _ o ypww nvmy qvo pzuto
pn |mrqupro pr vuk x{ wmm¦prt uq qvo zul ®¯ ur x{ prqkmj|prt
u vmzmwmt{ o|mzlmnpqpmr m} _
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o ypww jno qvo njxtkmjl o|mzlmnpqpmr un mjk vmzmwmt{ o|mzlmnpqpmr
yvp|v |ur xo on|kpxo pr qvo }mwwmyprt zurrok pknq woq : xo u |mwwo|qpmr m}
njxtkmjln m} |wmno jrok |mr; jtuqpmr_ oq_ <) = xo qvo : mkxpq |uqotmk{
>
yvp|v pn qvo |uqotmk{ ypqv mx; o|qn }mk : ur ypqv zuln un
qvo zmklvpnzn_ oq xo qvo pr|wjnpmr }jr|qmk }kmz <= qm qvo |uqotmk{ m}
nlu|on_ mzlmnprt ypqv qvo 'mkow |mrnqkj|qpmr tpson u }jr|qmk
= <= _ mqp|o qvuq = vun qvo vmzmqml{ q{lo m} }mk ur{ njxtkmjl _ vpn }jr|qmk ruqjkuww{ prj|on urmqvok }jr|qmk
= vm|mwpz = qvoko}mko = tpson u vmzmwmt{ o|mzlmnpqpmr m} mrw{ p} : pn ur uzlwo |mwwo|qpmr m} njxtkmjln m} noo p} ur
q qvpn lmprq ur uzlwo |mwwo|qpmr m} njxtkmjln m} ypww xo _pn|jnno
nqukqprt ypqv qvkoo o~rpqpmrn_
xo u njxtkmjl _ pn nup qm xo mk
nqjxxmkr p} vun rm rmrqkpspuw rmkzuw njxtkmjln_
" oq
"" oq
xo u njxtkmjl _
pn u {wmy njxtkmjl m} _

pn nup qm xo p}
xo u |orqkp| njxtkmjl _ pn nup
"& noo ^^ oq
qm xo p} vun rm rmrqkpspuw rmkzuw
njxtkmjln
_
my woq qvo |mwwo|qpmr : xo qvo noq m} uww lkpr|pluw kup|uw njxtkmjln
m} _ kmuw vun nvmyr qvuq qvpn pn ur uzlwo |mwwo|qpmr ^^ uwwmyprt }mk
u vmzmwmt{ o|mzlmnpqpmr m} _ wnm pq nvmjw xo rmqo qvuq ur{ {wmy
lnjxtkmjl m} pn |mrqupro pr qvo |mwwo|qpmr : ur qvuq ur{ lkpr|pluw
kup|uw njxtkmjl m} pn ro|onnukpw{ kup|uw_
oq rmqp|o qvuq qvo zul ®¯ }u|qmkn qvkmjtv ur prsokno wpzpq |mrnqkj|qo
spu qvo vmzmwmt{ o|mzlmnpqpmr on|kpxo uxmso
2 wpz
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2 ¥
F
zul pn prj|o x{ konqkp|qpmr ur qvo zul pn u lkm; o|qpmr mrqm
u volukqp|jwuk
owozorq
npr|o> : _ vo }mwwmyprt putkuz qvor |mzzjqon
²
yvoko qvo zuln ur uko lkm; o|qpmr mrqm qvo qvo noq m} otkoo |vuk
u|qokn m} ur mrqm qvo noq m} kolkonorquqpmrn m} konlo|qpsow{
!

2 ¡ 2 ¡ 2

wpz

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£
£
2

wpz ol
ol F
¡ ¡ 2

wpz vuk vuk ¥
F
mr qvo wo}q npo }mwwmy ounpw{ }kmz qvo oukwpok pn|jnnpmr m}
vo pnmzmklvpnzn

o
rmqp|o
qvuq qvo |mzlmnpqo m} qvo orqpko kptvq vur |mwjzr pn
tkmjln
_
±
qvo zul °* yvmno pzuto yo yurq qm ~r_ )kmz² qvo putkuz pq pn mxspmjn
qvuq yo |ur prnqou ~r qvo pzuto m} qvo zul _
r mkok qm wmm¦ |wmnok uq qvo pzuto m} qvo zul ² wlokprn }jnpmr qvomk{
ypww
xo pn|jnno_ r lukqp|jwuk qvo o~rpqpmr m} you¦ |mr; jtuqpmr }uzpw{
ur ur ouzlwo ypww xo tpsor_
" wlokpr ^ oq xo u ~rpqo tkmjl u lkpzo pspprt
% % ur u {wmy njxtkmjl m} _ noq m} lupkn yvoko
ur pn |uwwo u lkmspo qvuq
yvorosok ur uko njxnoqn m} ur ² }mk qvoko uko
owozorqn ² ² > > ¥¥¥ m} ur owozorqn ² > ¥¥¥ m} nj|v qvuq
^ ²> ² ²>
+ }mk ^ ur ur
© ² ²> ² ²> ² }mk ^ ^_
vo ouzlwo yo ypww xo jnprt m} u you¦ |mr; jtuqpmr }uzpw{ pn tpsor pr qvo

}mwwmyprt qvomkoz m} mwn|vzpq _ r mkok qm nquqo qvo qvomkoz yo roo
qm tpso qym zmko o~rpqpmrn_
" wlokpr ^ oq ur xo {wmy njxtkmjln m} u ~rpqo
tkmjl ur woq _ pn nup qm xo u p} ur uko {wmy njxtkmjln m} _
" ~rpqo tkmjl pn |uwwo p} pq |mrquprn u lkmlok
njxtkmjl nj|v qvuq p} %% % ur }mk ur{ %% ² %_
r qvpn |uno pn |uwwo u nqkmrtw{ ozxoo njxtkmjl m} _
" mwn|vzpq ^ vomkoz ©_ # % % $ # ^ + 2 © * $
$
vo roq lkmlmnpqpmr ypww xotpr qm kowuqo wlokprn }jnpmr qvomk{ ypqv qvo

prsokno wpzpq qvuq vun² xoor jno pr qvpn lulok _ vpn ypww uwwmy jn qm ~r qvo
pzuto m} qvo zul _
"
# % % $ $
kmm} o ypww nvmy qvuq pn u you¦ |mr; jtuqpmr }uzpw{ x{ nvmyprt
qvuq un u noq pq |mrquprn qvo you¦ |mr; jtuqpmr }uzpw{ m} mwn|vzpq _
_ o ~knq rmqp|o qvuq pn u |orqkp| njxtkmjl m}
oqnpr|o
2 pn u {wmy njxtkmjl m} x{ qvo }u|q qvuq pn u
quzo prqokno|qpmr_ pr|o pn u {wmy njxtkmjl m} * ur pn |orqkp| zjnq uwnm xo lkpr|pluw kup|uw noo ^^ ozuk¦ ^_^+_ " # % % $ : vuk vuk ² Fwpz
vuk vuk $
kmm} )pknq yo ypww nvmy qvuq qvo pzuto m} ² pn |mrqupro pr vuk _
jllmno qvuq wpz vuk _ ªo~ro u zul }mk ou|v : un
Fprsokno
wpzpq m} qvo |mzlwo |vuku|qok m} _ mqp|o
qvo lkm²; o|qpmr }kmz qvo
oq
o~ro
qvuq 2 _ ur woq zour 2 _ oq xo qvo
noq |mrnpnqprt m} uww lupkn yvoko : ur _ '{ qvo
wunq lkmlmnpqpmr pn u you¦ |mr; jtuqpmr }uzpw{_
jllmno qvuq nj|v qvuq ypqv ² _ '{ qvo
o~rpqpmr m} you¦ |mr; jtuqpmr }uzpw{ qvoko opnqn ² > ¥¥¥ : ypqv
}mk ^ ur ² > ¥¥¥ nj|v qvuq
²> ² ²>
}mk ^
ur
² ²> ² ²> ² }mk ^ ^_
yo rmqp|o qvuq % konlo|qn }jnpmr pr ur
konlo|qn }jnpmr pr _ )kmz qvpn yo noo qvuq qvo }mwwmyprt qvkoo
nquqozorqn vmw
²²
²
²> ² ²> ²> ² ² ²> ²
}mk ou|v ^ ² ^ ur
²> ²> _
jqqprt qvono nquqozorqn qmtoqvok yo noo qvuq qvoko}mko konlo|qn }jnpmr pr ² ur nm pn |mrqupro pr vuk _
mro qm mro |mkkonlmror|o pq pn ormjtv qm nvmy qvo
m nvmym}qvuq
pn u zullprt
opnqor|o
ur prsokno
2 vuk wpz vuk ¥
F
oq vuk _ o ypww o~ro x{ tpsprt qvo |vuku|qokn yvp|v
ypww xo ²2 }mk ou|v : _ )2p ² : _ voko opnqn nj|v
qvuq _ m woq _ pr|o konlo|qn }jnpmr pr
pq pn oun{ qm noo qvuq qvpn o~rpqpmr m} 2 pn yoww o~ro_ vpn
o~rpqpmr qvor tpson qvo }mwwmyprt o~rpqpmr m} 2
2 wpz ¥
F
q pn mxspmjn }kmz qvo o~rpqpmr qvuq
² 2 pn qvo porqpq{ zullprt _ o
²
rmy vuso qm nvmy mrw{ qvuq 2 pn qvo porqpq{ zullprt _ oq xo ur
owozorq m} qvo prsokno wpzpq ur woq 2 _ psor : mxnokso qvuq
zjnq xo o~ro uxmso x{ qvo ruqjko m} qvo : mkxpq |uqotmk{ <= _
nvmyn qvuq 2 ² }mk ou|v : qvoko}mko
vpn mxnoksuqpmr
2 ² zjnq xo qvo porqpq{ zullprt _ _
llw{prt vomkoz +_¬ nvmyn qvuq qvo zul ² pn ur pnmzmklvpnz yvor qvo
kurto pn konqkp|qo qm vuk _ nprt qvpn konjwq yo noo qvuq yo |ur jno
qvo zuln ±* ± ur prnqou m} qvo |mkkonlmrprt zuln ±°* ±° ur
° _ o uwnm noo qvuq p} qvo zul pn u zmrmzmklvpnz mk ur olpzmklvpnz
nm pn qvo zul ±* _ r mkok qm nqj{ qvo zul yo zjnq ullw{ mxnqkj|qpmr
qvomk{_
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o|uww qvo }jr|qmk = <= yvp|v yo or|mjrqoko
pr o|qpmr +_

o ypww rmy |mzlmno = ypqv qvo }jr|qmk mz qm toq u }jr|qmk
= <= yvp|v qu¦on qm u nlu|o ypqv qvo vmzmqml{ q{lo m}
mz _ o ypww noo qvuq qvo mxnqkj|qpmrn qm qvo zul
xoprt
u zmrmzmklvpnz mk ur olpzmklvpnz wpo pr qvo jllok wpzpqn wpz = _
)mk ou|v ^
qvo mxnqkj|qpmrn wpo yo ypww roo qm ullw{ qvo |mnpzlwp|puw ko
m noo yvoko
lwu|ozorq
}jr|qmk _ voko uko ruqjkuw pnmzmklvpnzn = ! =
}mk ou|v ur wpz = ! = }mk ou|v ¯ |vulqok _
vono qmtoqvok tpso qvo ruqjkuw pnmzmklvpnz wpz = ! = }mk
ou|v
¯ _

r owozorq wpz
|ur xo qvmjtvq m} un ur owozorq
F
m} mq ² = _ vpr¦prt m} pq un ur owozorq m} mq² = qvo mx
pzuto m} qvo zul wpo pr qvo |mvmzmqml{
nqkj|qpmrnqm² xoprt prqvo
²
!
tkmjln
=
= }mk ^_ pzpwukw{ qvo mxnqkj|qpmrn qm
vusprt u jrpjo lkopzuto pr wpo pr qvo |mvmzmqml{ tkmjln
= !
= }mk ^_
o|uww qvuq }mk ur{ kolkonorquqpmr ypqv u tkmjl
= ! noo ¬_
pr|o qvo |orqkuwpok m} u ~rpqo njxtkmjl m} pn qvo lkmj|q m} sukpmjn
pq pn |wouk qvuq pn npzlw{ |mrro|qo
qvoko}mko
> ² = ² ² = ¥
vo mxnqkj|qpmrn yo roo qm prsonqptuqo qvor wpo pr qvo |mvmzmqml{ tkmjln
= wpz =
}mk ^ +_ r lukqp|jwuk p} wpz² = }mk + qvor qvo
zul pn ur olpzmklvpnz ur p} wpz = }mk + qvor qvo zul
pn ur zmrmzmklvpnz _ o ypww roq pn|jnn npqjuqpmrn yvoko qvono vptvok
wpzpqn uko proo qkpspuw_
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'o}mko yo lkm|oo pr nvmyprt yvor qvono vptvok wpzpqn surpnv yo ~knq
roo u |mjlwo m} o~rpqpmrn_ vono ypww uwwmy jn qm jno nquruk qvomk{ ~knq
prqkmj|o x{ §u|¦myn¦p ¨|wjko ur iwpsok ^+ ^© _
}jr|qmk <= pn |uwwo ur }jr|qmk p} pq
pnmzmklvpnz |wunn m} mx; o|qn r uqmzp|
surpnvon mr uww xjq lmnnpxw{ u nprtwo
}jr|qmk ypww xo nup qm xo |mr|orqkuqo mr ur pnmzmklvpnz |wunn_mr yvp|v pq
mon rmq surpnv_
" ªo~rpqpmr _ m} ^© )mk ur{ lkpzo ~rpqo tkmjl ur
zmjwo woq <2 xo qvo uqmzp| }jr|qmk

|mr|orqkuqo mr qvo }koo mkxpq ^ ypqv ^ _ oq
wpz _
vo }mwwmyprt wozzu }mwwmyn pko|qw{ }kmz ymk¦ x{ §u|¦myn¦p ¨|wjko
ur iwpsok pr lukqp|jwuk }kmz qvo ~knq lkmm} m} ozzu _ pr ^+ _
& : <2 wpz ! wpz % $
< =

:$ wpz ! ¥
o ypww xo ~wqokprt mjk }jr|qmk x{ uqmzp| }jr|qmkn pr mkok qm noo yvoko
surpnvon _ vo kowosurq wozzu pn u konjwq m} ymk¦ x{ kmuw ^^ _
oo uwnm ^© kml _ _^^ ur kmlmnpqpmr _ _
oo ^^ # $ k¦ ² %% % $ $
ozzu _ tpson jn qvo }mwwmyprt lkmlmnpqpmr yvor u }jr|qmk pn ~wqoko

x{ uqmzp| }jr|qmkn_
# % %$ # : $ : k¦ <= wpz $
nprt qvpn lkmlmnpqpmr pr |mr; jr|qpmr ypqv qvo ymk¦ m} qvo lkospmjn no|
qpmr yo toq qvo }mwwmyprt konjwq _
# % %$
$ # : : k¦ + $
: k¦ ^
$
r qvo konq m} qvpn lulok yo ypww jno vomkoz _ ur sukpmjn konjwqn uxmjq

~rpqo tkmjln qm lkmso qvo qvkoo qvomkozn lkonorqo pr qvo prqkmj|qpmr_
_ 9hahb3 cdf45 46cb `
o ypww nqukq x{ lkmsprt vomkoz ^_^_ o xotpr qvpn lkm|onn ypqv qvo
}mwwmyprt wozzu yvp|v nvmyn qvuq qvo mxnqkj|qpmrn surpnv yvor u {wmy
njxtkmjl m} vun u |orqok m} nzuww pro pr pqnow} _
# % %
$ # $
% $
$
kmm} oq
xo u |orqkp| njxtkmjl yvp|v pn rmq u {wmy

njxtkmjl m} _ o zu{ unnjzo yvp|v pzlwpon qvuq _ oq yvp|v pn u {wmy njxtkmjl m} _
qvuq npr|o pn |orqkp|
q pn |wouk qvuq yvp|v pzlwpon
qvoko}mko _ jllmno qvuq qvor
|mrqkup|qprt xoprt|orqkp|_ m yo noo qvuq pn nqkp|qw{ |mrqupro pr
my x{ v{lmqvonpn qvjn x{ qvo nqkp|q |mrquprzorq
_
_ vpn |woukw{ pzlwpon % _ vor npr|o
%
pspon
_

kmm} m} vomkoz ^ ^ n yun pn|jnno oukwpok pq pn ormjtv qm nvmy qvuq
qvuq _pspon
% % ± pn ur pnmzmklvpnz o unnjzo qvuq
}mk ou|v lkpzo
_
*
mon rmq |mrqupr u kur¦ qym owozorquk{
uxowpur njxtkmjl
oq xo
_

u {wmy njxtkmjl m} } pn |{|wp| pq pn uxowpur qvoko}mko qvo mrw{
|orqkp| njxtkmjln m} _uko qvo {wmy njxtkmjln_ '{ vomkoz _ qvo
zul pn ur pnmzmklvpnz un pn qvo zul ±* _ ir qvo mqvok vur p} pn
rmq u |{|wp| tkmjl qvor + ur pn u torokuwpo juqokrpmr tkmjl _
jllmno pr qvpn |uno qvuq pn u |orqkp| njxtkmjl m} _ o noo qvuq
opqvok pn qvo |{|wp| tkmjl m} pro qym pr pn |{|wp| m} mkok }mjk ur
prqokno|qn qvo |{|wp| tkmjl m} pro qym pr pr u |{|wp| tkmjl m} mkok qym
mk pqnow} pn u torokuwpo juqokrpmr tkmjl _ } pn m} pro qym pr
% _ r qvo no|mr |uno pn u juqokrpmr tkmjl m} mkok
2 o juqokrpmr ur rmq u {wmy
optvq ur % _ } pn torokuwp
njxtkmjl pq }mwwmyn qvuq 2 pn u torokuwpo juqokrpmr tkmjl ypqv
2 + yvpwo p} pn u {wmy njxtkmjl 2 ^_ o
rmy lkmsor pr ou|v |uno qvuq p} pn u |orqkp| njxtkmjl m} qvor
vuso
>
%
}jkqvokzmko qvo zul pn ur pnmzmklvpnz_ )kmz mjk
lkospmjn pn|jnnpmr yo noo qvuq qvo zul ±* pn uwnm ur pnmzmklvpnz_ vo lkmm} m} vomkoz ^_+ }mwwmyn }kmz qvo roq wozzu_
" oo ^ _©_^ # $ %% %
k¦ + $
kmm} oq %% % ur
xo u lkpr|pluw kup|uw njxtkmjl m} nj|v
qvuq k¦ +_ jllmno pn rmq u {wmy njxtkmjl m} _ oq
_ vo okpso njxtkmjl m} pn u njxtkmjl noo
^ _©_^_ )mk u njxtkmjl m} yo ypww ormqo x{ ° qvo pz
uto m} pr _ my npr|o pn lkpr|pluw kup|uw ° _
vpn pzlwpon ° _ m ° pn ur uxowpur tkmjl _
pr|o
pn rmq u {wmy njxtkmjl m} % _ pn lkpr|pluw
kup|uw qvoko}mko pn u {wmy njxtkmjl m} _ vpn nvmyn
qvuq % qvjn vun u rmrqkpspuw rmkzuw
njxtkmjl _ vpn |mrqkup|qn xoprt u lkpr|pluw kup|uw njxtkmjl m}
nm pn u {wmy njxtkmjl m} _ ||mkprt qm ozzu p} pn u kur¦ qym tkmjl m} m mkok ypqv %% %
qvor qvo zul pn ur pnmzmklvpnz ur qvo zul ±* pn uwnm ur pnmzmk
lvpnz_ q }mwwmyn }kmz qvo prqkmj|qpmr qvuq p} pn u kur¦ qym tkmjl m} m
mkok qvor qvo zul ± pn uwnm ur pnmzmklvpnz qvjn lkmsprt vomkoz ^_+_
vpn mrw{ wouson vomkoz ^_© qm xo lkmsor_ n yo vuso xoor mprt pr
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A QUOTIENT OF THE SET [BG,BU(n)] FOR A FINITE GROUP G OF

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