Is a Christian woman who was married to a Muslim man, then deserted her husband, and was later informed by him of her divorce without being given a divorce certificate, considered lawfully divorced based on his statement, and is it permissible to marry her?

The fundamental principle is that the marriage remains valid unless a divorce is proven, either by the husband's confession or by evidence. A woman whose husband's divorce has not been proven is not permitted to marry another. If the husband denies divorcing her, she is not permitted to allow him access to herself and should avoid him. She should not marry until her divorce is evident. Divorce by writing does not take effect unless intended by the husband. It is advisable to consult the Sharia court in such cases involving claims, and it is recommended to seek a righteous Muslim woman.