Research Collection
Doctoral Thesis
On the cohomology of finite groups with simple coefficients Author(s): Stricker, Markus Andreas Publication Date: 1989 Permanent Link: https://doi.org/10.3929/ethz-a-000518488
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ETH Library
Diss. ETH Nr. 8887
On the
Groups
Cohomology
with
of Finite
Simple Coefficients
A dissertation submitted to the
Swiss Federal Institute for the
Technology Zürich
of
degree
of
Doctor of Mathematics
Presented
by
Markus Andreas Stricker
dipl. Math. ETH born September 6, 1960 Citizen of Zürich and Grabs
Accepted
on
Prof. Dr. U.
Prof. Dr. G.
(SG)
the recommendation of
Stammbach, examiner
Baumslag,
co-examiner
Prof. Dr. U. Stgmmbjch Math*-?!.*' tiK ETH-Zentrum HG G65.2
j-8002
k% 1989
Zijrich ^
Abstract
Let the order of the finite group G be divisible
be
a
field of characteristic p.
procedure is
no
to
such
compute the
procedure
It is well-known that there is
cohomology
to concoct the
intersections of centralizers of carry
of
some
simple
minimal
projective
fcG-modules with
simple
centralizers, termed C'(G), will be studied in this
G, CX(G)
group
Although no
systematic more
is laid
on
simple
groups
a
=
0^P{G)
and also
general and practica! because there
modules,
resolution of k.
non-vanishing
Obviously,
i-th
the
cohomology
C'(G)
the
that,
by
of
C'(G)
One of the
p-solvable
an
arbitrary
p-solvable, C*(G)
Orrp(G)
in
=
finite
OjT{G).
general,
there
of this thesis is to
objectives
subgroups C'(G). Special emphasis
groups of
p-length
1 and extensions of
p-groups.
is either
of the above mentioned result
p-nilpotent
It is well-known that for
a
Hn'G/Opi(G),M). Hence, Op/(G).
For
problem
to decide whether
an
for
result that for
when G is
need not coincide with
account in the literature.
analysis
paper.
surprising
Information about these characteristic
generalization
G, C2(G)
the rather
proved
it is well-known that
obtain
As
no
coefficient
Information about the minimal projective resolution of fc. These intersections
Griess and Schmid have
is
with
p and let k
by the prime number
arbitrary
or
not
group or
not
prove that for
an
arbitrary
group
p-solvable.
fcG-module M in the we can
we
principal block
restrict ourselves to the
G, C'(C'(G))
C'{H)
=
H,
=
for
C'(G).
of
G, Hn(G,M)
study of Hence,
subgroups
H of
=
groups with trivial
it is a
an
given
important
group. One
of
our
main results deals with
Let G
(Pi
=
x
Furthermore
•
•
¦
x
Pj)
a
X
we assume
special
2 ist und mindestens eine
p-Gruppen
(1
/)
< »
3
Theorems ist G
projektiven Auflösung werden
C'(G) explizit
bestimmen.
speziellen
Klasse
C2(G)
Als Versuch
sind und
speziellen
Q
eine
ein Normalteiler
von
für
p-Gruppe
beliebige Gruppen
man aus
C2(Q)
schliessen kann.
zu
und
n
(Cp
x
•
¦
¦
x
1.
Mit Hilfe einer
G alle
Untergruppen
in der
Analyse
1 sein.
bestimmen, diskutieren
C2(G)
l)
i
1.
Cp) XiQ.
Gruppen
< »
—>
G
ist. Für die meisten endlichen nicht-abelschen einfachen
über die =
Q
mit
für alle
gilt C"(G) ^ G,
wir für solche
Kranzprodukten
pen und für die meisten Primzahlen p
Voraussetzungen
=
P, XlQ (1
=
Gruppen G, (1
Dies wiederum wird das
endliche nicht-abelsche einfache Gruppen wobei P eine
Gruppe G,
p-überauflosbar ist, dann gilt Cn(G) / G,
überauflösbar sind und {
ob
einer
1:
ist. Weiter nehmen wir an, dass P,
nicht
einer
p-Länge
von
G ist.
von
Ein
handelt
Hauptresultate
gilt C2(Q)
Erweiterung,
e, ohne
liefert
Voraussetzungen
=
e.
C2{G)
=
über die
Dies, P.
zusammen
Q,
Grup¬
mit weiteren
Es ist eine offene
Erweiterung,
für
—»
auf
Frage,
C2(G)
=
P
