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Let A be a graded, commutative Hopf algebra. We study an action of the symmetric group Σ n on the tensor product of n − 1 copies of A; this action was introduced by the second author in  and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory  . We show that for a certain class of Hopf algebras the cohomology ring H * (Σ n ; A ⊗n−1 ) is independent of the coproduct provided n and (n − 2)! are invertible in the ground ring. Then, by choosing adoi:10.1017/s0013091599000826 fatcat:hrblpxgmq5ba3mj3sgw5qgurgy