I've taken 2 (undergrad) courses in abstract algebra and a reading course in Galois Theory, and I still don't understand the point of studying groups and rings. The courses have not been particularly difficult for me, but my motivation is extremely low. In Galois Theory obviously I saw an application of groups to solve a problem. And I can follow the proofs. But I still don't understand the connection between groups and solvability by radicals in any intuitive sense. And I haven't gained any insight into how to apply groups to solve other problems. This is completely different from analysis, where it is extremely clear to me how it can be used to solve various problems. And also in analysis, I have enough intuition and insight about it that I can prove theorems on my own readily. How do people develop intuition with groups and rings? Can you develop a visual or geometric or physical sense about them, or is it forever just symbols on a page that you manipulate by the rules? Do people actually enjoy this? What other problems can groups and rings solve? Thanks.