Is it permissible for a woman to travel without a mahram for the purpose of studying or visiting relatives, given the availability of safety in transportation and modern means of communication?

The basic ruling is that it is impermissible for a woman to travel without a mahram, based on the Prophet's (peace and blessings be upon him) saying: "It is not lawful for a woman who believes in Allah and the Last Day to travel for a day's journey except with a mahram." An exception to this is made in cases of extreme necessity.