90
IX. TH E TRAC E FOR M I N DIMENSIO N 7
EXAMPLE
33.27 . Amon g th e subgroup s o f od d inde x o f A7 , there ar e th e fol -
lowing:
SL2(Fg) = Ae, SL.2(F7) , Qi6 , 54 , S5 .
(Here S
n
denote s th e uniqu e centra l extensio n o f S
n
b y {±1} i n whic h th e trans -
positions an d th e produc t o f two disjoin t transposition s lif t a s elements o f order 4 .
This i s not th e sam e a s the S
n
o f 25.3.)
33.28.
ANOTHERJDEFINITIO N O F TH E INVARIAN T
e . Th e definitio n o f e ca n
be extende d t o an y A
n
(n 6 ) a s follows : Conside r th e uni t quadrati c for m
qn = (1,1,..., 1) o f ran k n an d th e correspondin g orthogona l grou p O
n
. Ther e i s
a natura l embeddin g o f S
n
i n O n(fc), whic h map s A
n
i n SO n(fc). On e check s (se e
e.g. [Se84 , 2.3] ) tha t A
n
i s contained i n th e imag e o f the coverin g ma p
Spinn(fc) - + SO n(k)
and tha t it s invers e imag e i n Spin
n
(/c) i s A
n
. W e thus ge t a n embeddin g
(33.29) An - » Spin
n
(k)
(which i s in fac t th e constructio n o f A
n
originall y give n i n [Schu r 11] for k C).
Hence an y continuou s homomorphis m
ip:
T
k
- A
n
defines a n elemen t (p) o f H l(k, Spin
n
). B y usin g th e Araso n invarian t ( a specia l
case o f Rost' s invariant )
#H^Spin
n
)--+iJ
3
(/c),
we map ((f) t o a n elemen t o f H 3(k). W e thus ge t a normalized invarian t e
n
o n th e
An-torsors.
(Here is a more direct description of the invariant. Fo r t an element of
H1
(fc , A n),
let E(t) b e th e correspondin g ran k n etal e algebr a an d writ e q
t
fo r it s trac e form .
By 33.2 0 an d 33.21, w e hav e wi(q t) = 0 an d W2(qt) = 0 . Thi s mean s tha t th e
element q
t
qn o f the Wit t rin g W(k) belong s t o J 3, wher e / i s the augmentatio n
ideal of W(k). Th e Araso n ma p J 3 // 4 H 3(k) transform s q
t
qn int o an elemen t
of H 3(k), whic h i s easily see n t o equa l e
n
(t).)
When n 5, the invarian t e
n
i s 0, see Prop. 32.1.
When n = 6 , 7, one checks that e
n
coincide s with the invarian t e defined above .
For n 8, it i s no longer tru e tha t e
n
i s unramified, a s simple example s show ;
hence, on e canno t us e i t t o settl e Noether' s proble m fo r A n.
34. Furthe r application s t o Noether' s Proble m
34.1. Se t H t o b e the quaternio n grou p Qi 6 o f order 16, i.e., th e grou p gener -
ated b y tw o elements r , s wit h th e defininin g relation s
r8 = 1, s 2 = r 4, an d srs -1 = r _ 1.
The cente r Z(H) o f H i s {1,
r4
}. W e identify th e grou p H/Z(H) wit h th e dihedra l
group D4 a s follows : vie w D4 a s th e subgrou p o f S4 which stabilize s th e partitio n
{1,2}, {3,4} , an d ma p r t o th e cycli c permutatio n (13 24) an d s t o (12). Th e
quadratic characte r x o f i f define d b y H D4 S4 {1,-1 } i s give n b y
x(r) = - 1 an d x(s ) = - 1 .
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