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Let k = Fp be the field with p > 0 elements, and let G be a finite group. By
exhibiting an E∞–operad action on Hom(P,k) for a complete projective resolution P
of the trivial kG–module k, we obtain power operations of Dyer–Lashof type on Tate
cohomology Ĥ*(G;k). Our operations agree with the usual Steenrod operations on
ordinary cohomology H*(G). We show that they are compatible (in a suitable sense)
with products of groups, and (in certain cases) with the Evens norm map.
These theorems provide tools for explicit computations of the operations for
small groups G. We also show that the operations in negative degree are
nontrivial.
As an application, we prove that at the prime 2 these operations can be used to
determine whether a Tate cohomology class is productive (in the sense of Carlson)
or not.
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