Is it permissible for a divorced woman to go out with her son without the knowledge of his father? Can she move to her own home, and if so, when? Can the father prevent her from moving with her son?

A woman is permitted to go out for any permissible (mubah) purpose according to Islamic law without her father's permission, provided she adheres to the Shari'ah guidelines. She is also allowed to reside wherever she wishes if she is safe from temptation (fitna) and free from suspicion. The default is for a woman to remain in her home, as Allah Almighty says: "And abide in your houses." Your son does not need to be with you for you to reside alone in a dwelling. However, it is preferable to live with the father if possible.