Basically, no permit is required for children traveling alone in Germany. If your child travels abroad on their own or with friends, or even with an organization (such as a sports club or language study trip), then in some countries it is advisable or even compulsory to bring a parental consent or authorization (https://www.adac.de/-/media/adac/pdf/jze/vollmacht-allein-reisendes-kind.pdf?la=de-de&hash=ED158FCD4C32394889253E13829B3C3115EBF859).