Is it permissible to chat with a female friend online without the knowledge of her mother, who has generally forbidden her from communicating online, given that the relationship does not cause harm to any party?

A child must obey their parents in matters that benefit them and do not harm the child. The mother's fear regarding her daughter's communication through chat rooms is justified due to the potential for evil and temptation within them. The daughter must obey her mother in this regard, whether in her mother's presence or absence, and whether her mother is aware or not. The mother's prohibition does not mean she views her daughter's friend as unsuitable; rather, it is because she is unaware of her circumstances, and the prohibition applies generally to all those whose circumstances are unknown to her. If your friend can convince her mother to agree, then praise be to Allah; otherwise, let her go and communicate with righteous women.