Someone is in an abusive marriage as the husband has a habit of beating her in front of her children and giving really low and bad words to her and her family members. Is it permissible for him to do so and if not then what should she be doing?

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It is not permissible for the husband to hit his wife or abuse her with insults and abuse, whether for her or her family, and if he did, he is sinning, and he is exposed to the wrath of God Almighty, and if the beating causes effects on the wife’s body, then he must pay the blood money for her, according to the effect that occurred.
As for what can be do, she can refer the matter to the legal ruler (S. Sistani in your case), and he will order him to live with her in a kind manner or to divorce her, and if he refuses, the ruler will divorce her from him.