Some time ago my mother found out my father was cheating on her. He apologized and she forgave him and we expected the other woman to realize and leave him too but she didn't. She insisted with him to divorce my mother, leave me and my brother, sell his home and live with her and marry her despite that woman herself being a married woman. My father apologized to us but some time later we found out again that he was still cheating… My mother wanted to ask another important thing. Is it permissible for my mother as wife to file for divorce?

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If he has been abusive to her, she can seek a Khul’i divorce. However, that has to be handled in a religious manner by contacting a local mosque that offers divorce services. Once the Islamic divorce is done, she can go to the court and file a legal or civil divorce.