I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear algebra, but the students have already had a semester of linear algebra so I'm thinking about how to deal with group theory.
I'd like to present some examples of applications of group theory that motivate the theoretical questions the course will deal with. For instance, "symmetries of three-dimensional objects form groups. Crystals have symmetrical structure. If we could get some bounds on possible forms of three-dimensional symmetry, this would give limits on the sorts of crystals that could form, which would be interesting to a chemist." Another example could be, "The roots of a polynomial can sometimes be permuted in ways that do not change the value of polynomial functions of the roots, and these permutations form a group. [insert explanation of Galois theory here] If we could determine whether this group is trivial, we could see whether solving the polynomial is possible."
My goal is to persuade the students that group theory is useful and therefore interesting, but since they will most likely be interested in different things, I'd like to have a big list in the hope of being able to find something for each of them. Can people suggest either applications of group theory or places to find such applications?