Is it permissible for a woman to travel without a mahram, and if it is not, how do we respond to those who cite the concept of safe companionship to permit it?

A woman's travel without a mahram (unmarriageable kin) is forbidden, and there are various prophetic narrations prohibiting it, among them the saying of the Prophet, peace be upon him: "It is not permissible for a woman who believes in Allah and the Last Day to travel for a day and a night's journey without a mahram accompanying her." Those who permit her travel without a mahram in safe company restrict this to obligatory Hajj and Umrah, and to cases of compelling necessity. As for other journeys, some scholars have mentioned a consensus on their impermissibility for a woman without a mahram.