Do Women Really Make Less Than Men?

I am confused. I hear over and over about women making less money than men for the same work. For years the woman has been suppressed by this lower pay rate. Well how's this:

If women make less than men, then it is cheaper for a woman to do the work.

If it's cheaper for a woman to do the work, the employer saves money by women doing the work.

If the employer could save money, why don't employers hire all women?

I need help with this. Please post your thoughts.