I understand that a woman can divorce her husband. But why can't she divorce him by saying 3 times divorce (talaq).
Many non muslim say it is oppression of woman. And it is gender inequalty.
The man is to pay the woman he marries a mahr. If she is later divorced just because the man wanted to replace her, she is not left with nothing.
“But if you want to replace one wife with another and you have given one of them a great amount [in gifts], do not take [back] from it anything. Would you take it in injustice and manifest sin?”(Qur'an 4:20 translation)
Ibn Kathir, may Allah have mercy on him, said: “This means: if any one of you wants to divorce his wife and replace her with another, you should not take anything from what you have already given the first wife, even if it was a huge amount of wealth.(source of translation: islamqa.info)
Narrated Ibn `Umar:
The Prophet (sallallahu alayhi wa sallam) said to those who were involved in a case of Li’an, "Your accounts are with Allah. One of you two is a liar. You (husband) have right on her (wife)." The husband said, "My money, O Allah's Apostle!" The Prophet(sallallahu alayhi wa sallam) said, "You are not entitled to take back any money. If you have told the truth, the Mahr that you paid, was for having sexual relations with her lawfully; and if you are a liar, then you are less entitled to get it back." (source of translation: sunnah.com)
Men do not have the right to mahr like women do.
There are two types of separation known as talaq and khula’. Talaq is given by the man and khula’ is initiated by the woman. There are significant differences in their rulings.
Some texts on talaq:
“Divorce is twice. Then, either keep [her] in an acceptable manner or release [her] with good treatment. And it is not lawful for you to take anything of what you have given them unless both fear that they will not be able to keep [within] the limits of Allah . But if you fear that they will not keep [within] the limits of Allah, then there is no blame upon either of them concerning that by which she ransoms herself. These are the limits of Allah, so do not transgress them. And whoever transgresses the limits of Allah - it is those who are the wrongdoers.“(2:229 translation)
فَإِن طَلَّقَهَا فَلَا تَحِلُّ لَهُۥ مِنۢ بَعْدُ حَتَّىٰ تَنكِحَ زَوْجًا غَيْرَهُۥ ۗ فَإِن طَلَّقَهَا فَلَا جُنَاحَ عَلَيْهِمَآ أَن يَتَرَاجَعَآ إِن ظَنَّآ أَن يُقِيمَا حُدُودَ ٱللَّهِ ۗ وَتِلْكَ حُدُودُ ٱللَّهِ يُبَيِّنُهَا لِقَوْمٍۢ يَعْلَمُونَ
“And if he has divorced her [for the third time], then she is not lawful to him afterward until [after] she marries a husband other than him.1 And if he [i.e., the latter husband] divorces her [or dies], there is no blame upon them [i.e., the woman and her former husband] for returning to each other if they think that they can keep [within] the limits of Allah. These are the limits of Allah, which He makes clear to a people who know [i.e.,understand].” Saheeh International
It was narrated from ‘Alqamah from Ibn Mas’ood (may Allaah be pleased with him) that he was asked about a man who married a woman but did not stipulate a dowry, and he did not consummate the marriage with her before he died. Ibn Mas’ood said: She should have a dowry like that of her peers, without exaggeration or falling short, and she has to observe the ‘iddah, and the inheritance is hers. Ma’qil ibn Sinaan al-Ashja’i stood up and said: The Messenger of Allaah (peace and blessings of Allaah be upon him) passed a ruling concerning Birwa’ bint Waashiq – one of our womenfolk – similar to what you have said. Ibn Mas’ood rejoiced at that. (source of translation: islamqa.info)
Ibn Qudaamah said: “The disliked divorce is to divorce without any need for it. Al-Qaadhi said, "There are two scholarly views on this. The first, it is prohibited because the husband wrongs himself and his wife and it constitutes denying both of them the benefits of marriage without real need; therefore, it is prohibited just like unjustifiably destroying wealth. Moreover, the Prophet(sallallahu alayhi wa sallam) said: 'There should be neither harm nor reciprocal harm.'"” [Al-Mughni] (source of translation: sunnah.com)
According to scholars, if talaq is only given once, or twice, the woman is still entitled to maintenance. Thus, she is not entirely separated from the husband until he divorces her three times.
Khula’ causes separation just by happening once.
Some texts on khula’:
The Hanbali scholar Al-Buhooti wrote, "If the husband neglects any of the rights of Allah, it is recommended for the wife – just as it is recommended for the husband in a similar situation – to request termination of the marriage through divorce or Khulʻ for his neglect of the rights of Allah upon him." (source of translation: islamqa.info)
وَحَدَّثَنِي عَنْ مَالِكٍ، عَنْ نَافِعٍ، عَنْ مَوْلاَةٍ، لِصَفِيَّةَ بِنْتِ أَبِي عُبَيْدٍ . أَنَّهَا اخْتَلَعَتْ مِنْ زَوْجِهَا بِكُلِّ شَىْءٍ لَهَا فَلَمْ يُنْكِرْ ذَلِكَ عَبْدُ اللَّهِ بْنُ عُمَرَ . قَالَ مَالِكٌ فِي الْمُفْتَدِيَةِ الَّتِي تَفْتَدِي مِنْ زَوْجِهَا أَنَّهُ إِذَا عُلِمَ أَنَّ زَوْجَهَا أَضَرَّ بِهَا وَضَيَّقَ عَلَيْهَا وَعُلِمَ أَنَّهُ ظَالِمٌ لَهَا مَضَى الطَّلاَقُ وَرَدَّ عَلَيْهَا مَالَهَا . قَالَ فَهَذَا الَّذِي كُنْتُ أَسْمَعُ وَالَّذِي عَلَيْهِ أَمْرُ النَّاسِ عِنْدَنَا . قَالَ مَالِكٌ لاَ بَأْسَ بِأَنْ تَفْتَدِيَ الْمَرْأَةُ مِنْ زَوْجِهَا بِأَكْثَرَ مِمَّا أَعْطَاهَا .
Yahya related to me from Malik from Nafi from a mawla of Safiyya bint Abi Ubayd that she gave all that she possessed to her husband as compensation for her divorce from him, and Abdullah ibn Umar did not disapprove of that.(source of translation: sunnah.com)
Malik said that divorce was ratified for a woman who ransomed herself from her husband, when it was known that her husband was detrimental to her and was oppressive for her, and it was known that he wronged her, and he had to return her property to her. Malik added, "This is what I have heard, and it is what is done among us."(source of translation: sunnah.com)
From Muwatta Malik, Book 29, Hadith 32
It was narrated from Ibn 'Abbas that the wife of Thabit bin Qais came to the Prophet and said: "O Messenger of Allah, I do not find any fault with Thabit bin Qais regarding his attitude or religious commitment, but I hate Kufr after becoming Muslim." The Messenger of Allah(sallallahu alayhi wa sallam) said: "Will you give him back his garden?" She said: "Yes." The Messenger of Allah(sallallahu alayhi wa sallam) said: "Take back the garden and divorce her once."(source of translation: sunnah.com)
Grade: Sahih (Darussalam)
Disclaimer: As with all my answers, I am not giving a fatwa. I share what I know of Islamic scripture, historical events and words of scholars, and translations of them, and they may or may not be the correct answer to the question, based on whether I interpreted them correctly as being relevant or not.