Is it permissible for the wife to request a divorce upon discovering her husband's betrayal through phone calls and emails, his flirtation with another woman, and their exchange of pictures, love, and romantic messages, even though she has not been neglectful towards him?

If the husband is in an illicit relationship with another woman, he is disobedient to Allah and committing a reprehensible act. The wife should remind him of the gravity of sins.

If he does not respond and insists, then she has the right to seek a divorce. However, if he repents and ends the relationship, she does not have the right to seek a divorce from him unless she is harmed by remaining with him. Divorce is not a trivial matter, especially with the presence of children, and one should not resort to it except when all means of reconciliation have failed.