  The question whether global symmetries can be realized in
quantum-gravity-matter-systems has far-reaching phenomenological consequences.
Here, we collect evidence that within an asymptotically safe context, discrete
global symmetries of the form $\mathbb{Z}_n$, $n>4$, cannot be realized in a
near-perturbative regime. In contrast, an effective-field-theory approach to
quantum gravity might feature such symmetries, providing a mechanism to
generate mass hierarchies in the infrared without the need for additional
fine-tuning.
