88 IX. TH E TRAC E FOR M I N DIMENSIO N 7
PROOF. Th e invariant b is clearly normalized. Le t us show that i t is unramified .
Let k b e a n extensio n o f fco which i s complet e fo r a discret e valuatio n whic h i s
trivial o n /co , and le t I denot e th e correspondin g inerti a subgrou p o f IV Le t x b e
an elemen t o f i/^fc , G) ; cal l p
x
th e correspondin g homomorphis m Tk G. W e
have to show that b{x) = (2 ) -d(x) i s an unramified elemen t o f H2(k). I f (fx(I)7 ^ G,
the cohomolog y clas s d(x) i s unramified, an d s o is b(x). I f p x{I) G, th e residu e
field contains th e 2
m-th
root s o f unity; sinc e m 3, this implie s tha t 2 is a squar e
in th e residu e field, henc e 2 i s a squar e i n k an d b(x) = 0 . Thi s show s tha t th e
invariant b is unramified ove r ho-
Take k' t o b e th e unramifie d extensio n o f degre e 2
m
o f k = Q 2 (2-adi c field).
Then k'/k define s a n element x o f
i/1(/c,
G) such that d(x) i s equal to 5 (in k*/k*
2).
Since th e quaternio n /c-algebr a (2,5 ) i s a divisio n algebra , w e have (2 ) -(5) ^ 0 in
H2(k),
whic h shows that b(x) i s not 0. Henc e b is a nonzero element of
IIIVQ(G,
Z/2Z )

THEOREM
33.16. Let G be a group with a 2-Sylow subgroup which is cyclic of
order 8. Then Rat(G/Q ) and a fortiori Noe(G/Q ) are false.
PROOF.
Le t H denot e a 2-Sylo w subgrou p o f G . Sinc e i t i s cyclic , i t ha s a
normal complemen t N [Sco87 , 6.2.11] . Th e invarian t b define d abov e give s a n
invariant a of G via th e ma p G H. Thi s invarian t i s normalized an d unramifie d
since b is. Le t k b e a n extensio n o f Q an d le t t b e a n elemen t o f if
1^,
H) suc h
that b(t) i s nonzero. Sinc e th e ma p i7
1
(fc, G)— H
x(k,H)
i s surjective , a i s not 0
and w e may appl y 33.11.
REMARK
33.17. Th e specia l cas e o f Th . 33.16 wher e G i s abelia n wa s prove d
by:
Endo-Miyat a [E M 73], Voskresenski i [V o 73], an d Lenstr a [L e 74] (fo r th e
Noe property) ;
Voskresenski i [V o 77] and Saltma n [Sa l 82] (fo r th e Ra t property) .
Saltman's proo f use s a cohomological invarian t wit h value s i n H
2(k)
whic h first
appearances t o th e contrar y i s the sam e a s th e invarian t b defined i n Exampl e
33.14 above .
33.18.
TH E GROUPS ^
A
n
AN D TH E CORRESPONDIN G ETAL E ALGEBRAS .
Fo r
every n 4 , w e writ e A
n
fo r th e invers e imag e o f A
n
i n th e extensio n S
n
» S
n
from 25.3 ; this is the unique non-trivial extension of An b y {±1}. W e have an exac t
sequence
(33.19) 1 - {±1} - T
n
- A
n
- 1.
Consider no w a n etal e algebr a E o f rank n ove r a field /c, and le t
WE
'• r ^ S
n
be th e correspondin g homomorphis m a s i n 3.2 . Le t Wi(qE) b e th e Stiefel-Whitne y
classes o f the trac e for m qE- B y 25. 3 and 25.10, we have:
(33.20) Th e image of (fE i s contained i n A
n
i f and onl y if w\{qE) = 0 .
Suppose tha t th e imag e o f pE i s containe d i n A n. Th e ho -
(33.21) momorphis m (fE : T^— A
n
ca n b e lifte d t o A
n
i f and onl y if
w2{qE) = 0 .
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