Women In Ancient Greece

How did women's roles in society differ from Athens and Sparta?
Women's roles in Athens were that they would always serve the men. They weren't that high up in class, and unlike Spartan women they weren't expected to learn how to read or write. Spartan women had more of a larger role and weren't so low down in class. Women in Athens were only a little bit higher in class compared to the slaves.
If the Greeks are considered so advanced, so culturally elite, why do you believe gender equality (equality between men and women) was not established?
I think that since things were mainly run by men and women didn't really have a say. Men in that era were just probably seen as being more dominant so they saw it as that the men should always decide what would be going on.