Should a child be made to write a letter of apology without a parents consent beforehand?

My daughter (10 yrs old) has been experiencing problems at school with her friend. Over the past few weeks her friend has been pushing her around and then refusing to talk to her and if my daughter is playing with somebody else she then pushes her out of the way. I told my daughter to tell a teacher but she said that she did not feel she could speak to them. I then told her to ask her friend why she was being like this and to try and talk it out with her. Anyway this went on for a couple of weeks and this week her friend once again pushed her so my daughter retaliated and pushed her back. I told her this was wrong and said that she should have just walked away. Anyway the girl went home to her mother and said that she felt very threatened and frightened by my daughter. Her mother then wrote to the teacher telling her what her daughter had said. My daughter was spoken to about the incident and told that the headmaster had been informed. However, she was not asked how it had all come about and was then told to write an apology letter to the girl. I woudl just like to add that my daughter did try to apologise to the girl after she had done it but the girl just kept ignoring her. When my daughter gave the girl the letter she just looked at it and threw it in the bin. She then told people that my daughter had pushed her, hurt her back and then hit her. My daughter denies this but does admit to pushing her. I want to know if the teacher was allowed to make my daughter write the letter without my consent. Any help woudl be greatly appreciated. I was also told by the teacher that before my daughter had written the letter they had in fact made up.