Is educating a girl child permitted in Islam? If so, then what are the rules?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

The education of girls is as compulsory as the education of boys. Therefore, it is compulsory to give all children (males and females) the obligatory education, i.e. teaching them the matters that are compulsory on them and the matters which they are forbidden from doing.  In addition, it is desirable to teach them what is recommended in the Sharee'ah and what is reprehensible as to let them know how to act and behave in the right way.

Also, we should teach children the profession that suits their nature. Males should be taught what they need in their life and females should be taught what fits their nature. The Prophet used to fix a special day to teach women their religious matters apart from men.

However, teaching women should be free from any kind of prohibitions, such as mixing with males or not covering their 'Awrah, etc.

Allaah Knows best.