'Authority' in the Bible

Nadab and Abihu died in the LORD's presence when they offered unauthorized fire before him in the Sinai wilderness. Since they didn't have their own children, Eleazar and Ithamar ministered as priests under the authority of Aaron their father.

This is the law in cases of jealousy when a woman defiles herself while under her husband's authority:

including anything that the LORD commanded you by the authority of Moses, starting from the day the LORD commanded Moses and continuing through your generations.

to serve as a memorial to the Israelis, a reminder that no unauthorized person, who isn't a descendant of Aaron, is to attempt to burn incense in the LORD'S presence, so that he may not become like Korah and his group, just as the LORD had spoken by the authority of Moses.

turning over your authority to him so that the entire community of Israel knows to obey him.

laid his hands on him, and charged him, just as the LORD had commanded, using Moses' authority.

and told him, "Your servants took a count of the soldiers who were under our authority. We didn't miss a single man.

Here's the travel itinerary for the Israelis after they left the land of Egypt in groups under the authority of Moses and Aaron.