Does a divorce take place when a husband says to his wife: "You are divorced," after her false confession of having an affair with another man, followed by her regret and request to return, and is it permissible to take her back?

If there was originally no intention of divorce, and the divorce occurred based on a lie, then in this case, the divorce does not take place, as some jurists have stipulated, citing fatwas from Sheikh Muhammad ibn Ibrahim Al ash-Sheikh and Ibn al-Qayyim. However, if the divorce occurred based on what was attributed to the wife, whether it was proven or not, then it does take place. Marital life must be based on trust and good assumptions between the spouses, in accordance with His (Allah's) saying: {O YOU WHO HAVE BELIEVED, AVOID MUCH [NEGATIVE] ASSUMPTION. INDEED SOME ASSUMPTION IS SINFUL.} And if the wife lied or had bad assumptions, then she must repent and be cautious of anything that weakens the marital bond.