There are, and have been what is called 'Matrilineal' societies, as for example, among the Nairs of Kerala and the Khasis of Meghalaya, but though it is true that the line of inheritance of the family's property, and descent, was from mother to daughter among them, coupled with the custom of a woman continuing to be a part of her maternal household even after marriage, and her spouse being a visiting husband, the power and management of the family property was very much in the hands of men.

The only difference was that here, it was the maternal uncle who was the ruler and caretaker of his sisters' children, instead of the father, who enjoyed the same powers in his maternal home. Therefore it was not really a female dominated society, but an egalitarian society. The main disadvantage of this system, according to what I have heard, is the laxity of marital ties because of the married couples not living together. Otherwise, it was a better social system, meting out less cruelty to women than the patriarchal system.

In any case, the rising incidence of men burning their wives for dowry, and raping their own daughters has belied the image of men being the protectors of their wives and daughters. They are more likely to abuse their proprietary rights over them by raping or murdering them, or selling them for immoral traffic, instead of cherishing and guarding them.

In view of the evils of patriarchy, I often wonder which social system would be the better option to replace patriarchy. As mentioned before, there was a female-oriented (not female-dominated ) social system called matrilineal society in ancient Kerala and Meghalaya, probably descended from Arabs and Babylonians who followed 'Mother Right.'

However, though this system invested women with a higher status and more security, it became untenable in course of time with men having to leave their native place and settle in far-away lands for earning a better living, as it couldn't cope with the need of men to meet their wives and family regularly, which led to strained and fickle relations between married couples.

The third  option is the Communal system, namely the Kibbutz in Israel, wherein people live in mutually cooperative groups or Communes, comparable with our 'Ashrams', which I think is a good option in that there must be more support of society to women and children.

The last option is the loose live- in relationship system without the bonds or shackles of marriage as the people in the Western countries choose to call it and follow.

The choice is a tad difficult and it will not be surprising if the ultimate outcome is a mixture or amalgam with three or four types of systems coexisting and accommodating all the  above options in some degree, in a symbiotic relationship, with people free to choose their own utopia.