TITLE: what are the different applications of group theory in CS?
QUESTION [41 upvotes]: What are some applications of abstract algebra in computer science an undergraduate could begin exploring after a first course?
Gallian's text goes into Hamming distance, coding theory, etc., I vaguely recall seeing discussions of abstract algebra in theory of computation / automata theory but what else? I'm not familiar with any applications past this.
Bonus: What are some textbooks / resources that one could learn about said applications?

REPLY [2 votes]: The Walsh-Hadamard code's robustness relies on the subset-sum principle which is most easily seen as a consequence of the second fundamental theorem of linear algebra after viewing inner products as linear maps. This code and its robustness properties is used in the proof of the PCP theorem which is used to prove hardness of approximation of MAX3SAT and establishing hardness results elsewhere in the study of approximation algorithms.
Computation of the permanent, which is a function analogous to the determinant, is a complete problem for $\mathbf{PP}$. 
The complexity theoretic classes $\mathbf P$ and $\mathbf{NP}$ are defined as algorithms that run in deterministic resp. nondeterministic polynomial time. Robustness of these classes relies on closure properties of polynomials; for instance, the property that the sum of any two polynomials is again a polynomial.
Infinite families of hash functions are constructed by considering finite fields, a familiar object from abstract algebra. They have many applications including a parallelizable derandomization of the randomized algorithm for MAXCUT and the proof that $\mathbf{BPP} \subset \mathbf{PH}$.
The proof that $\mathbf{IP} = \mathbf{PSPACE}$ relies on arithmetization of the operations $\wedge$, $\vee$, $\neg$, $\forall$ and $\exists$ to convert SAT formulae into polynomials, which are often studied in algebra. Important properties of polynomials such as bounding the number of roots by the degree are considered in this proof.