Is it permissible for a mother to transfer ownership of her apartment to her son, and then for him to buy another apartment for his sisters at a lower price, and will the half-brother inherit from this apartment, knowing that the father had transferred ownership of the apartment to the mother and she paid half its price from her own money?

The default is the obligation of justice among children in giving, and this includes the giving of both the father and the mother. If all adult, mature children consent to the gift, then it is permissible. If the sisters' intention is a complete waiver of their right to the apartment for their brother, then there is no harm. However, if the registration is for a purpose other than a complete waiver, with the intention that the apartment is theirs, then it is not advisable unless there is something that guarantees the sisters' rights. A husband may give his wife whatever he wishes from his wealth, and it becomes her property, inherited from her. The husband's son does not inherit from the wife unless there is another reason for inheritance.