1. Even though many students take a course is discrete math where they study various proof techniques many of them seem
not to absorb this material well. Abstract algebra provides them much more practice at this in a different context than
discrete math does.
2. High school math teachers should be very adept at modular arithmetic. Cyclic groups is where they learn this well.
3. Group theory is the mathematics of symmetry--a fundamental notion in science, math and engineering. For example, the symmetry group of a molecule reveals some of its possible (or impossible) chemical properties.
4. There are many important practical applications of modular arithmetic that are best understood by viewing the modular arithmetic in a group theory framework. Examples include the check digits on UPC codes on retail items, ISBN numbers on books, and credit card numbers. In many cases the check digit is the inverse of a weighted sum modulo an integer (10 in the case of a UPC number, 11 in the case of an ISBN number, 9 in the case of Visa travelers checks).
5. Many games can be understood by viewing them as permutation groups. Two examples are the 15 puzzle and the Rubik cube.
6. High school math teachers should be adept at looking at data and making plausible conjectures and generalizations. They should also teach their students to do this. This is a skill that can be learned with practice. Groups and rings provide abundant opportunities for developing this skill.
7. Many people are not comfortable with abstract concepts nor adept at abstract reasoning. The ability to think abstractly is a valuable asset. Abstract algebra helps develop this ability.
8. Abstract Algebra is an ideal capstone course for math ed majors and for those who will go on to grad school in math. Throughout the course they review things like 1-1 functions, onto functions (surprisingly few senior math ed majors understand these ideas well); equivalence relations; basic concepts from linear algebra such as how to multiply matrices, properties of determinants, how to compute a determinant, how to compute the inverse of a matrix, how to tell if a matrix has an inverse, linear transformations (which are group homomorphisms); properties of complex numbers; properties of integers (Euclid's lemma, division algorithm, criterion for divisibility by 9 or 11 or 4); math induction (another important topic that many students do not seem to understand well when they begin an abstract algebra course--this is especially the case for statements that do not involve sums of series); and properties of polynomials (division algorithm, remainder theorem, factor theorem, number of zeros is at most the degree, unique factorization).
9. Doing well in an abstract algebra course is a confidence builder and sometimes causes students to think about going on the graduate school. I once had a student who did extremely well in abstract algebra who went to medical school and now has a high position in the Center for Disease Control in Atlanta. About 20 years after she took the course I met her for dinner while I was at a meeting in Atlanta. I jokingly said to her "Did you use any abstract algebra in med school?" She immediately responded by saying "Abstract algebra was very valuable to me in med school." I asked how. She said that whenever she was taking a difficult course she said to herself "If I can get an A in abstract algebra I can get an A is any course." She was perfectly serious. Many people who start out intending to be high school math teachers or even teach high school math for several years decide to go to grad school in math for an advanced degree (I and many others loved the course and wanted to go to grad school to continue studying the subject). Taking abstract algebra and doing well makes such a move more likely and easier to do.