I know that polygamy is permissible in Islam.
But I have heard that if a husband had agreed (before they got married) that he would not marry a second woman, while in marriage with her, the husband is obliged not to do that.
So, does this mean it is permissible if prior to marriage, the woman asked the man not to marry another woman while she's able to carry out her duty as a wife?
This condition is permissible, and that must be fulfilled..
God say:
O YOU WHO believe, fulfil your obligations. [5:1]
And the Prophet upon him said:
The stipulations most entitled to be abided by are those with which you are given the right to enjoy the (women's) private parts (i.e. the stipulations of the marriage contract). Sahih al-Bukhari 5151
And as did the Prophet Muhammad, upon him with his daughter Fatima, when he conditioned Ali not to marry the other, and asked him to divorce her if he married other
"If a woman does not feel that she could allow her husband to marry more than one woman at the same time, then Islam allows her the right to refuse him permission to do this at the outset of their marriage, however, she must indicate this preference in the Prenuptial Agreement or she will forfeit this right under the Islamic Law. If she is uncertain as to whether or not she will be opposed to her husband marrying a second wife later on, then she could include that in the agreement and thus make it binding upon her husband that he must consult her at that time and that he must then abide by her wishes. To say nothing, however, could possibly invite more pain than gain as far as her desires are concerned. In the West, polygamy is illegal. Even so, the woman may still request that her husband not marry a second wife, and put this in the contract. This sort of request would be considered spurious in the Prenuptial Contract because men in the west are already forbidden polygyny. Nonetheless it might still prove to be a useful addition to the contract at this time because later on the couple might possibly move to a country where polygyny is legal."
You can look more in here
There is and other view, it says:
"because of the harm and mess that inevitably results:
a) the harm to the first wife;
b) the troubles with the second wife when the first is upset;
c) the harm of not giving both their legal, emotional, and material rights;
d) the harm to family relations;
e) and, also vitally, the harm to one’s children…"
You can find resource in here
Allah Most High says:
“If you fear that you shall not be able to deal justly with the orphans, marry women of your choice, two, three, or four, but if you fear that you shall not be able to deal justly (with them), then only one…(Surah al-Nisa, 3)."
So at this conditions,in my opinion; if first wife refuse to allow, husband can't have permission.
