Essential cohomology and relative cohomology of finite groups

Abstract

In this thesis, we study mod-p essential cohomology of finite p-groups. One of the most important problems on essential cohomology of finite p-groups is finding a group theoretic characterization of p-groups whose essential cohomology is non-zero. 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 p-group. To determine the relative essential cohomology, we calculate the essential cohomology of an elementary abelian p-group. 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 p-groups. Finally, we define inflated essential cohomology and in the case p > 2, we prove that for non-abelian p-groups 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].