The image of the transfer map
D. J. Benson
Mathematical Institute
24-29 St. Giles
Oxford OX1 3LB
Great Britain
1 Intro duction
In this paper, we prove a generalization of a theorem of Carlson [1]:
Theorem 1.1 Suppose that G is a finite group, and k is a field of characteris*
*tic p. Let
H be a collection of subgroups of G. Denote by Kthe collection of subgroups K o*
*f Gwith
the property that the Sylow p-subgroups of the centralizer CG(K) are not conjug*
*ate to a
subgroup of any of the groups in H.
Let J be the ideal in H (G; k) given by the sum of the images of transfer fr*
*om subgroups
in H, and J0be the ideal in H (G;k) given by the intersection of thepkernels!of*
*prestriction!
to subgroups in K. Then the ideals J and J0have the same radical, J = J0.
!
!
!! The theorem of Carlson is the special case of the above theorem in which H i*
*s taken
to!be the collection of subgroups whose indexis divisible by p. In this case, *
*Kis the
collection!of!subgroups whose centralizer contains some Sylow p-subgroup of G, *
*and J 0is
the!kernel!of restriction to the center of a Sylow p-subgroup.
!
!
Theorem!1.2 (Carlson) Suppose that G is a finite group,and k is a field of ch*
*aracter-
istic!p.!Let J be the ideal in H (G;k) spanned by the transfers from subgroups *
*of index
divisiblepby!p,pand!J0the!kernel!of restriction to the center of a Sylow p-subg*
*roup. Then
!!J = J0.
!
!! Our proof is a generalization of a simplified version of the proof of Carlso*
*n's theorem
given!by!Evens and Feshbach [3].
!
The author is very grateful for the support andhospitality of the Sonderfors*
*chungsbereich 170 "Ge-
ometrie und Analysis," Universit{t G|ttingen, for the period during which this *
*research was carried out.