Most of the modern theory of error-correcting codes has its roots in abstract algebra. It's also important in theory - Babai's recent celebrated quasipolynomial

Most of the modern theory of error-correcting codes has its roots in abstract algebra. It's also important in theory - Babai's recent celebrated quasipolynomial algorithm for graph isomorphism relies on group theory. Most of the recent results in cryptographic obfuscation also rely on group theory because they encode circuits as permutation matrices using Barrington's theorem.