Is it permissible to divorce one's wife after discovering her infidelity, taking into account the presence of children and family relations, and how should one act regarding this?

A husband discovering his wife's betrayal is a tremendous tribulation and an evil deed, as it is a dereliction of God's right and an aggression against His right. The wife is obligated to guard her husband regarding herself and his property. If she has erred, then she must be advised and called to sincere repentance. If her repentance is sincere and her conduct improves, then he should keep her and live with her kindly. If she does not repent, then divorce is preferable. If she is divorced, he should not mention the true reason for the divorce to anyone. If the neighbor and his wife conspired to corrupt the wife, then moving away from their neighborhood is better.