The power operation in the Galois cohomology of a reductive group over a number field

For a connected reductive group G over a local or global field K, we define a *diamond* (or *power*) operation (ξ, n)↦ ξⁿ\, \ H¹(K,G)× Z→ H¹(K,G) of raising to power n in the Galois cohomology pointed set (this operation is new when K is a number field). We show that this operation has many functorial properties. When G is a torus, the ponted set H¹(K,G) has a natural group structure, and our new operation coincides with the usual power operation (ξ,n)↦ ξⁿ. For a cohomology class ξ in H¹(K,G), we define the period per(ξ) to be the greatest common divisor of n≥ 1 such that ξⁿ=1, and the index ind(ξ) to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting ξ. We show that per(ξ) divides ind(ξ), but they need not be equal. However, per(ξ) and ind(ξ) have the same set of prime factors.