The computation of transfer maps, evens norm maps and Steenrod operations

The mod-p cohomology of any group is a module over the steenrod algebra, hence it is naturally that this algebra plays a great important role in the theory of group cohomology. This dissertation describes the method and its implementation in the computer algebra system Sage for computing the Steenrod Operations on the mod-2 cohomology rings of finite groups. We often us the detection method to relate the Steenrod operations on the cohomology rings of a given groupto the ones of its subgroups. In case the detection methos does not work, we shall try the computations using combinatorially the inflation, the transfer map and the comodule map. There are some certain cases that the above methods are not enough but the order of group is appropriate, then we shall use the Evens norm map to generate the Steenrod operations. In the dissertation, we also discribe our implementations to calculate the transfer and the Evens norm map. We obtain from the first run of our Sage package the Steenrod operations on the mod-2 cohomology ring of the Mathieu group M22, and 49 groups of the 51 groups of order 32. The remaining cases of the groups of 32 will be completed soon.


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