What is the ruling regarding a husband's oath to divorce his wife if she cheats on him or if he looks at another woman with lust? And does the divorce take place without the wife's knowledge? And what is the ruling regarding swearing on the Quran?

If by betrayal is meant adultery, then it is a grave sin that necessitates immediate repentance. He must conceal his sin and not inform anyone of his transgression.

A conditional divorce takes effect if the condition upon which it is suspended is met, whether the wife knows or not.

The husband must abstain from sins and lower his gaze. Spouses should not use the suspension of divorce as a means to solve problems; rather, they should use advice and admonition.