Is it permissible for a man to marry his thrice-divorced wife after she married another man with the consent of both parties, with one guardian and one witness, and without an offer or acceptance, then that man divorced her after consummating the marriage, knowing that a فاسد (invalid) marriage does not make her lawful for her first husband?

The 'ijab (offer) is for the guardian to say, "I marry my daughter to you," and the qabul (acceptance) is for the suitor to say, "I accept." The marriage contract is concluded by anything that indicates it customarily, such as saying "I give my daughter to you in marriage" or "I grant you possession of my daughter," and this is the view of Sheikh al-Islam Ibn Taymiyyah.

However, if the contract is concluded without any verbal pronouncement, such as merely signing a paper, it is void, and the contract must be re-established with verbal 'ijab and qabul.

As for witnesses, the majority of scholars stipulate them, while the Malikis permit delaying them until before consummation. Some scholars, including Sheikh al-Islam Ibn Taymiyyah, are of the opinion that public announcement suffices instead of witnesses. So, if the marriage is publicly announced, it is valid. However, if both witnessing and announcement are absent, the contract is invalid by consensus.

If the contract is void due to the absence of verbal 'ijab and qabul, or due to the absence of witnessing or public announcement, then this marriage does not make the woman lawful for her first husband who divorced her three times; because a فاسد (corrupt/invalid) marriage is not a true marriage.