I know that during a girl's period she is not allowed to touch the Quran.
Can she read Quran without touching it, like reciting, or reading from the PC's screen, or mobile?
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AFAIK a muslim (man and woman) is allowed to recite Quran from memory - independent of the state he is in... although there are some opinions stating that if men are in the state of janaabah they need to first make ghusl.
Regarding "women during period" reading it from a screen/mobile etc. there seems to be no "100% consensus"... some say you are allowed, others say you are not... for a detailed analysis see here and here.
Allah knows best.
in the Hanefi mezheb(most comman concept), you can not recite or read without touching it.It is just like men's impure state, us (men) we can not read or recite Qur'an when we are in that situation. We have to immediately take the ablution. But you can read blessing(words not in the Qur'an).
Qur'an 56:79 (none touch it except the purified) implies that a person in an impure state shouldn't touch the Qur'an◊, which includes women during menses. If she were to not touch the Qur'an, there's a difference of opinion as to whether she may read it, as described by IslamWeb:
The majority of scholars believe that it is unlawful for the woman, while in her menstruation period, to read the Qur'an. ...
But Imam Malik thinks that it is lawful for her to read it without touching it. This opinion was also reported from Imam Ahmad. It is also one of the opinions of Imam Al Shafi'e. ... We believe that this last opinion is the one most supported by evidence, as mentioned in your question.
Woman during menses handling Quran, IslamWeb
IslamWeb have other fatawa on this same topic: Woman during menses handling Quran (she may look to the Quran and read with her eyes only), and Touching Qur'an and praying during menses (she is allowed to read and review her memorization of the Qur'an without touching it).
Regarding reciting out loud, Islam Q&A states: The majority of fuqahaa’ say that it is haraam for a woman to recite Qur’aan during her period (and likewise by AskImam). See also: Can a woman recite the Qur'an during her menses?
This boils down to whether an electronically displayed Qur'an is mushaf and thus necessitates being in a state of wudu to handle. I found two major opinions on this:
They're not mushaf:
Mobile phones and other electronic devices in which the Quran is stored are not considered as a Mus-haf (Quran book) and the ruling of a Quran does not apply to them with regard to the obligation of being in a state of ablution to touch it and being disliked to enter the toilet with it and so forth.
They're effectively mushaf when Qur'an verses appear on the screen.
... once the verses of the Qur’an appear on the screen, it will not be permitted to touch the area where the actual text appears without Wudu. Touching other parts of the iPhone or device, however, will be permitted.
SeekersHub, presenting on viewpoint
◊ مجاهد's answer to Can Qur'an be touched without wudu? gives some nuance to this, but fatawa indicate it's usually considered impermissible, e.g., Islam Q&A writes It is not permissible for a Muslim to touch the Qur’aan when he does not have wudoo, according to the majority of scholars. Other example fatawa making this same claim are IslamWeb, SeekersHub, and albalagh.net.