I don't know about very specific applications, but I think the field itself and is very complimentary to CS, especially when considering finite or discrete structures. As it's name suggests, abstract algebra abstracts "nice" properties of, e.g., integers and formalizes them in a very concise and general manner. Modern abstract algebra is deeply tied to category theory, and so now these "nice" properties get abstracted even further out to maps between objects, and as maps between categories (i.e., functors). As such abstract algebra is tied to functional programming on some level (I know nothing about this connection though).

Linear algebra is a subfield of abstract algebra, and lots of general theorems about what classes of matrices are diagonalizable, or what their eigenvalues look like, etc. are within the purview of abstract algebra. These types of results are relevant to many algorithms, e.g., page rank.

Aside from that, I think abstract algebra is quite a beautiful field in its own right. Two books I would recommend are Artin's Abstract Algebra (as an intro) and Lang's Algebra (more advanced, good bridge into the category theory perspective).