"Under authority of" is a phrase that indicates something is happening with the permission and control of someone in a position of power. Some synonyms for this phrase include "by the order of," "with the approval of," "under the direction of," and "authorized by." These phrases all convey a similar meaning and can be used in a variety of contexts, such as legal documents, workplace policies, and government regulations. It is important to use these phrases accurately and appropriately to ensure that the level of authority is clearly understood and respected.