  Let k be the field with p>0 elements, and let G be a finite group. By
exhibiting an E-infinity-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 H*(G; k). Our operations agree with the
usual Steenrod operations on ordinary cohomology. 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 non-trivial. 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.
