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Bismihi ta'ala

In general, Islam prohibits abortion, and this is the primary law, considering abortion unlawful and a form of murder, where blood-money must be paid and a kaffarah.

In brief, there are certain circumstances where abortion becomes allowed. In the case of a rape incident, with the victim falling pregnant, before the fourth month (entering of the soul into the fetus), she is allowed to abort the fetus if there is extreme difficulty upon her to keep the baby. This means her mental state as the mother is unstable, her family and social status will be ruined, and she will not be able to bear the extremely damaging consequences of all of this. In this case, she is permitted to have an abortion. 

As for after the fourth month, there are different opinions on this, and she must refer to her Marja' taqleed for the fatwa. 

And Allah knows best.