What is the ruling on one who divorced, then took her back, then divorced, then took her back, then divorced a third time, and his wife did not know and he did not inform her of taking her back until after the third divorce?

If a man divorces his wife and then takes her back twice consecutively before the ‘iddah (waiting period) ends, and then divorces her a third time, she becomes irrevocably separated from him (baynunah kubra). She is not lawful for him until she marries another husband. The right of رجعة (taking back the wife) belongs to the husband, and it is not conditional upon informing the wife or having witnesses, though having witnesses is recommended.