Can a male and female betta fish be together in the same tank without a divider?

Yes, however if you were to do this than you'd want to make sure that they cannot see each other. Seeing each other would greatly stress out the fish and make them more susceptible to disease. Also, make sure the divider is very secure; if the divider was to slip up or out, and give them access to each other than the results would be disastrous.