Is it permissible for a woman to live alone, and if so, why is it not permissible for her to travel alone?

It is permissible for a woman to live alone if she is safe and not one of those accused of wrongdoing. However, she is not permitted to travel without a mahram, due to the Prophet, peace and blessings be upon him, saying: "A woman should not travel except with a mahram." This is because travel is a place of hardship and fatigue, and it exposes her to many dangers, just as it exposes her to temptation and enticement to evil. Travel is not to be compared to a woman residing alone in her own country, as the dangers in travel are greater than those in the country of residence.