Men have the Y chromosomes, women don’t. Women have no particular reason to preserve particular Ys because they don’t have any. If we want to preserve ours or increase their frequency, we have to do it ourselves.
But we want to preserve or increase our other chromosomes too. That means monopolizing access to the women who share our Xs and other chromosomes, or the alleles on those chromosomes. That implies exercising some level of power, control, or authority, over them.
Put another way, women and men BOTH have an incentive to prefer mating in-group (creating offspring that share more of our genes.) But men have a STRONGER incentive to mate ingroup, and to prevent women from mating out-group, than vice versa, because we have a whole chromosome they don’t have, but they don’t have any chromosomes that we don’t have.
Of course nobody “knows” (or nobody did until recently) what their genes and chromosomes are. But those tendencies and inclinations which have those effects will be evolutionarily stable, while others will be evolutionarily *unstable.* So genes which code for or assist those behaviors (ingroup mating preference, patriarchy) will tend to outcompete others.
Put another way, racism implies sexism. And racism is rational, so sexism is too.
Put another way, it’s naturally in our interests as men to rule women in their interests because their genome can be entirely a subset of ours, but it’s not necessarily in their interests to rule us in our interests, because some of our genome is necessarily not also theirs.
That’s one reason why men must rule.