Is a Woman Permitted to Live Alone in Islam?
Question

Can women live alone? If they can live alone, why can't they travel alone?
Praise be to Allah.Is a Woman Permitted to Live Alone in Islam?
A woman may live alone subject to the condition that she is trustworthy and is not a woman of dubious character. 
Can a woman travel without a Mahram?
With regard to a woman travelling without a Mahram , it is clearly not allowed, as in the Hadith narrated by Al-Bukhari (1729) and Muslim (2391) from Ibn `Abbas (may Allah be pleased with him) who said: The Prophet (peace and blessings of Allah be upon him) said: “No woman should travel except with a Mahram , and no man should enter upon her unless a Mahram of hers is present.” A man said: “O Messenger of Allah, I want to go out with such and such an army and my wife wants to go for Hajj .” He said: “Go out (for Hajj) with her.” 
Why can’t women travel without a Mahram?
This is completely wise, because travel involves hardship and exhaustion, and because of her weakness a woman needs someone to help her and stay by her side. Things may happen to her that may cause her to lose control when her Mahram is not with her. This is well known nowadays when there are so many accidents involving cars and other means of transportation. 
Moreover, if a woman travels alone, that exposes her to temptation and men may make approaches, especially when there is a great deal of corruption. Someone may sit near her who does not fear Allah, and he may tempt her to do something prohibited. 
If we assume that a woman is travelling alone in her car , she is exposed to other dangers, such as her car breaking down, or evil people ganging up on her, etc. 
This makes it clear that Islam is the first of all systems to take care of women, protect their honour, respect them and regard them as precious pearls that must be protected from evil. 
We submit to the command of Allah and His Messenger (peace and blessings of Allah be upon him), and we know that it contains complete wisdom and mercy, because Allah only forbids to His slaves that which is harmful for them. 
If a woman can live alone, why can’t she travel without a Mahram?
It is not correct to compare travel to a woman staying alone in a house in her own town, because there are more dangers in the place that a woman travels to. If a woman is in her own town, if anything happens to her or she needs someone to help her, she will find someone to help her. The fear of evil people attacking her is less when she is in her own town and her own house than when she is travelling. 
And Allah knows best.