A story I heard (perhaps enhanced over the years):
In 1910, Princeton was about to begin a major in physics. The physicists sat down to decide what would be required for those majors to study. As part of the exercise, they went through a list of mathematics courses to determine which would be used in physics. They came to "Group Theory", they all laughed. "There is a subject that will never be used in physics!"
But of course nowadays group theory is important in physics.*
Another story (you can probably find if you look). An interview with physicist Richard Feynman.
(discussion of group theory used in physics)
Interviewer: What would have happened if group theory had not already been developed by mathematicians?
Feynman: Some physicist would have taken a week off to develop group theory, then progress in physics would have continued.
*J. Cornwell, Group Theory in Physics, volumes I and II