15. STABILIT Y
37
R E M A R K 15.5. Th e stabilit y conditio n (15.2) involve s al l th e subgroup s H' o f
H. Ther e ar e specia l case s wher e on e ca n dispens e wit h mos t o f them. Fo r instance ,
let N b e a subgrou p o f G whic h contain s H an d whic h ha s "stron g control " o f th e
fusion o f H i n G , i n th e followin g sense :
/•p \ For every subgroup H 1 of H and every s G G with sH's~ x C H,
there exists t G N with txt~ l = sxs -1 for every x G H'.
P R O P O S I T I ON 15.6. Suppose that N normalizes H and has property (Fus )
above. Then an element of Inv(if , C) is stable if and only if it is fixed under the
action of N/H.
This follow s fro m Prop . 13.1.
E X A M P L E 15.7. I f H i s a p-Sylow o f G whic h i s abelian an d i V is its normalizer ,
a well-know n theore m o f Burnsid e say s tha t (Fus ) i s true . Le t u s recal l th e proof .
If s an d H' ar e a s i n (Fus) , le t G' b e th e centralize r o f H' i n G. Sinc e H i s abelia n
and contain s H\ w e hav e H C G'. Sinc e s~ 1 Hs D H\ w e als o hav e s~ 1 Hs C G'.
By th e conjugac y o f p-Sylows i n G' , ther e exist s c G G' wit h cHc' 1 s~1Hs. Th e
element t = sc ha s th e require d properties .
Hence, i f C i s a p-group , Inv(G , C) ma y b e identifie d wit h th e subgrou p o f
Inv(iJ, C) fixed unde r th e actio n o f N/H.
E X A M P L E 15.8. Take :
G = S
n
= grou p o f permutation s o f { 1 , . . . , n} ,
H = S
n
-\ stabilizer o f n ,
N = H = S
n
-L
Then (Fus ) i s true . Indeed , i f s an d H' ar e a s i n (Fus) , tak e t = s i f s(ri) = n.
If not , pu t m = s - 1( n ) , an d le t c G G b e th e transpositio n (run). Sinc e sH's~ x
is containe d i n H, i t fixes n , henc e H' fixes bot h m an d n. Thi s implie s tha t c
centralizes H f. On e the n put s t = sc . On e ha s t(n) s(m) = n , henc e £ belong s
to H an d i t i s clea r tha t txt~
x
= s x s
- 1
fo r ever y x E H
f
.
Hence ever y elemen t o f Inv(/S
n
_i,C) i s stable . I n particular , i f n i s prim e t o
|C|, £/i e restriction map Inv(S
n
,C)
Inv(Sf
n
_i,C) z s a n isomorphism. W e shal l
prove mor e precis e result s late r (25.14.2 an d 31.23). Indeed , w e shal l se e tha t th e
subgroup Inv n o r m (6'
n
, C) o f Inv(5
n
, C) i s 0 if \C\ i s odd, s o tha t th e interestin g cas e
is th e on e wher e n i s od d an d \C\ i s a powe r o f 2 , i n whic h cas e th e restrictio n ma p
Inv(5
n
,C ) Inv(S
n
-i,C) i s a n isomorphism .
P R O P O S I T I ON 15.9. Suppose H is a p-Sylow of G, and C is killed by a power
of p. Let a be an element of Inv(H,C). Assume:
(1) a is invariant under the automorphisms of H coming from the normalizer
N ofH in G ;
(2) Res f (a ) = 0 for every proper subgroup H' of H.
Then there is a unique element ac o/Inv(G,G ) such that R e s ^ a ^ ) = a ? namely
aG = Coi%(a)/(N : H).
P R O O F . Th e uniquenes s i s obvious , s o i t suffice s t o prov e existence . Defin e b
as Co r (a), an d comput e Res(6 ) b y th e doubl e cose t formul a (wit h H' = H). Th e
elements s G S whic h ar e no t i n N giv e 0 becaus e o f (2) ; thos e i n N giv e a becaus e
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