Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.