When is a woman required to be accompanied by a mahram, and when is it permissible for her to be alone?

Scholars have differed on the issue of a woman traveling without a mahram (definition), falling into two views: absolute prohibition except in cases of necessity (Hanafi and Hanbali schools), and permissibility only for the obligatory Hajj with a trustworthy companion (Maliki and Shafi'i schools). Ibn Taymiyyah chose the view that it is permissible for any journey of obedience with a trustworthy companion. Traveling by direct flight with a trustworthy companion is considered an instance of this, given the strong probability that she will not be harmed.