Yes I think the woman had to ask her husband for a bill of divorce. Unlike our legal system today, there were no "equal" rights between men and woman. The woman could not go to court to divorce her husband of her own will. I read something about once. So I guess you had situations where the woman wanted to divorce but couldn't, and the man refused to divorce, preferring to put her away and marry another (and thereby commit adultery, she was not properly divorced from the one he had put away), and men who would treat their wives badly , put them away, marry another, not get divorced as per the Law of Moses. The issue is a lot deeper than a simple "you can't remarry unless you commit adultery and here's the verse to prove it" tactic of today's christianity. Plus, we have a totally different legal system and rights about marriage, divorce etc than they did in Jesus's time. So it seems silly to me, to uphold ancient Jewish marriage and divorce laws on the basis of faulty or minimal understanding, in the context of our secular modern day legal system. We can't have it both ways.
Hmmmmm, that would make sense! Not actually divorcing but " putting away" and remarrying and committing adultery would make sense.