He divorced his Christian wife three times, then she married a Christian man, who then divorced her; is she permissible for her first husband?

Praise
be to Allah.

If a Muslim marries a Christian woman, then divorces her three times, then
she marries a Christian man after that in a valid marriage, and he is
intimate with her in the manner in which a man is intimate with his wife,
then he divorces her, in that case she becomes permissible for her first
husband, the Muslim, after the end of her ‘iddah. That is because of the
general meaning of the verse in which Allah, may He be exalted, says
(interpretation of the meaning): “And if he has divorced her (the third
time), then she is not lawful unto him thereafter until she has married
another husband. Then, if the other husband divorces her, it is no sin on
both of them that they reunite, provided they feel that they can keep the
limits ordained by Allah” [al-Baqarah 2:230]. And the Christian man in
this scenario is a husband. This is the view of the majority of scholars. 

Imam ash-Shaafa‘i (may Allah have mercy on him) said in al-Umm
(4/289): 

If
a Muslim man divorces his Christian wife three times, then a Christian or a
slave marries her, and has intercourse with her, then she becomes
permissible to the first husband if the second husband divorces her, once
her ‘iddah has ended, because either of these two men is a husband, and
Allah, may He be glorified and exalted, says (interpretation of the
meaning): “until she has married another husband” [al-Baqarah 2:230].
So in this case she has married another husband. End quote. 

Imam Ahmad (may Allah have mercy on him) was asked about a man who married a
Christian woman, then divorced her, then she married a Christian man who
then divorced her. Can she go back to the Muslim? He said: Yes; do you not
see that Allah says (interpretation of the meaning): “until she has
married another husband” [al-Baqarah 2:230]? And is this man not a
husband? End quote. 

And he was asked about a Christian woman who was married to a Muslim, who
divorced her three times, and her ‘iddah ended, then she married a Christian
man who consummated the marriage with her, then he died or divorced her; is
it permissible for her Muslim ex-husband to marry this Christian woman? 

He
said: Yes, because he was a husband. A Christian may make a dhimmi woman
permissible for a Muslim to remarry.

End quote from Ahkaam Ahl al-Milal wa’r-Riddah (p.170) 

It
says in al-Fataawa al-Hindiyyah (1/473): 

If
a Christian woman was married to a Muslim and he divorced her three times,
then she married a Christian man and he consummated the marriage with her,
she becomes permissible for the Muslim who previously divorced her three
times (to remarry). End quote. 

It
says in al-Mawsoo‘ah al-Fiqhiyyah (10/256): 

The majority of fuqaha’ are of the view that intercourse with her dhimmi
husband makes it permissible for her to remarry the first husband (if the
second husband then divorces her), because the Christian man is a husband.
But that is contrary to the view of Maalik, Rabee‘ah and Ibn al-Qaasim, who 
said that this does not make her permissile (for the Muslim man to remarry).
End quote. 

For more information, please see the answer to question no.
45645, 84849 and
82622. 

And Allah knows best.