Is it permissible for a father who bought a house and whose son contributed half its price to register only his own name in the purchase contract? And is it permissible for him to sell the house and take the money without giving his son anything? And is it permissible for the children to divide the house if the father passes away before them?

What the son paid to his father falls into one of three categories: Either it was a donation, in which case the son has no right to revoke it, and he becomes like any other heir in the inheritance upon the father's death. Or it was a loan, in which case it becomes a debt owed by the father that must be repaid, and the son has the right to recover his share from the inheritance before its division. Or it was a partnership in the house with the father's consent, in which case the house is jointly owned by them, and the father can only sell his share. If the father dies before that, half of the house remains the property of the son, and the heirs share in the father's half.