Essential cohomology and relative cohomology of finite groups
Author
Aksu, Fatma Altunbulak
Advisor
Yalçın, Ergün
Date
2009Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Show full item recordAbstract
In this thesis, we study modp essential cohomology of finite pgroups. One
of the most important problems on essential cohomology of finite pgroups is
finding a group theoretic characterization of pgroups whose essential cohomology
is nonzero. This is an open problem introduced in [22]. We relate this problem
to relative cohomology. Using relative cohomology with respect to the collection
of maximal subgroups of the group, we define relative essential cohomology. We
prove that the relative essential cohomology lies in the ideal generated by the
essential classes which are the inflations of the essential classes of an elementary
abelian pgroup.
To determine the relative essential cohomology, we calculate the essential
cohomology of an elementary abelian pgroup. We give a complete treatment
of the module structure of it over a certain polynomial subalgebra. Moreover
we determine the ideal structure completely. In [17], Carlson conjectures that
the essential cohomology of a finite group is finitely generated and is free over a
certain polynomial subalgebra. We also prove that Carlson’s conjecture is true
for elementary abelian pgroups.
Finally, we define inflated essential cohomology and in the case p > 2, we
prove that for nonabelian pgroups of exponent p, inflated essential cohomology
is zero. This also shows that for those groups, relative essential cohomology is
zero. This result gives a partial answer to a particular case of the open problem
in [22].
Keywords
Essential cohomologyinflated essential cohomology
relative cohomology
M`ui invariants
Steenrod algebra
Steenrod closedness
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