  For a non-elementary discrete isometry group $G$ of divergence type acting on
a proper geodesic $\delta$-hyperbolic space, we prove that its Patterson
measure is quasi-invariant under the normalizer of $G$. As applications of this
result, we have: (1) under a minor assumption, such a discrete group $G$ admits
no proper conjugation, that is, if the conjugate of $G$ is contained in $G$,
then it coincides with $G$; (2) the critical exponent of any non-elementary
normal subgroup of $G$ is strictly greater than half of that for $G$.
