Are there any applications of Abstract Algebra to Programming Language Theory? Is there anything that would be useful in language design and compiler implementation?
Algebra in the classical sense of the word is used in modeling of computational effects as algebraic operations, see for example these slides by Gordon Plotkin, or you can read real papers and Ph.D. theses, if you actually meant to ask the question.
I've hesitated to answer because any answer beyond a simple "yes" could and does fill volumes. Programming language semantics has been profoundly shaped by and in turn has deeply shaped the development of categorical logic, which is the application of algebra to logic.
But I suspect the best way of answering this question is to tell you to go learn Agda -- the experience of learning to program with dependent types is in large measure coming to grips with how comprehensively algebra pervades computer programming.
Actually there a software for theorical abstract algebra. This soft allow compute some kinds of things like actions groups, Homomorphism etc... It is Magma: http://magma.maths.usyd.edu.au/magma/
I hope this can help you.