well the title itself seems to be a paradox, but, What are some applications of abstract algebra (like groups, fields, and rings)? Apparently this determines the symmetry of particles in physics but what are some real-life, money-making application of group theory? (Yes, I money is one of my biggest concerns so I am also considering mathematical finance or acturial science. If I were to pursue pure math, number theory, analysis are still applicable to industry. But how about abstract algebra?