  For a connected reductive group $G$ over a local or global field $K$, we
define a *diamond* (or *power*) operation $$(\xi, n)\mapsto \xi^{\Diamond n}\,
\colon\ H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ 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 pointed set $H^1(K,G)$ has a natural group structure, and our new
operation coincides with the usual power operation $(\xi,n)\mapsto \xi^n$.
  For a cohomology class $\xi$ in $H^1(K,G)$, we define the period ${\rm
per}(\xi)$ to be the greatest common divisor of $n\ge 1$ such that
$\xi^{\Diamond n}=1$, and the index ${\rm ind}(\xi)$ to be the greatest common
divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting
$\xi$. We show that ${\rm per}(\xi)$ divides ${\rm ind}(\xi)$, but they need
not be equal. However, ${\rm per}(\xi)$ and ${\rm ind}(\xi)$ have the same set
of prime factors.
