I'm a computer science student among the things I'm learning Abstract Algebra, especially Group theory.
I'm programming for about 5 years and I've never used such things as I learn in Abstract Algebra.
what is the context between programming and abstract algebra? I really have to know.
Group theory is very important in cryptography, for instance, especially finite groups in asymmetric encryption schemes such as RSA and El Gamal. These use finite groups that are based on multiplication of integers. However, there are also other, less obvious kinds of groups that are applied in cryptography, such as elliptic curves.
Another application of group theory, or, to be more specific, finite fields, is checksums. The widely-used checksum mechanism CRC is based on modular arithmetic in the polynomial ring of the finite field GF(2).
Another more abstract application of group theory is in functional programming. In fact, all of these applications exist in any programming language, but functional programming languages, especially Haskell and Scala(z), embrace it by providing type classes for algebraic structures such as Monoids, Groups, Rings, Fields, Vector Spaces and so on. The advantage of this is, obviously, that functions and algorithms can be specified in a very generic, high level way.
On a meta level, I would also say that an understanding of basic mathematics such as this is essential for any computer scientist (not so much for a computer programmer, but for a computer scientist – definitely), as it shapes your entire way of thinking and is necessary for more advanced mathematics. If you want to do 3D graphics stuff or programme an industry robot, you will need Linear Algebra, and for Linear Algebra, you should know at least some Abstract Algebra.
I don't think there's any context between group theory and programming...or rather your question doesn't make any sense. There are applications of programming to algebra and vice versa but they are not intrinsically tied together or mutually beneficial to one another so to speak.
If you are a computer scientist trying to solve some fun abstract algebras problems, there are numerous enumeration and classification problems that could benefit from a computational approach to be worked on in geometric group theory which is a hot topic at the moment, here's a pretty comprehensive list of researchers and problems (of 3 years ago at least)
http://www.math.ucsb.edu/~jon.mccammond/geogrouptheory/people.html
popular problems include finitely presented groups, classification of transitive permutation groups, mobius functions, polycyclic generating systems
and these
http://en.wikipedia.org/wiki/Schreier–Sims_algorithm
http://en.wikipedia.org/wiki/Todd–Coxeter_algorithm
and a problem that gave me many sleepless nights
http://en.wikipedia.org/wiki/Word_problem_for_groups
Existing algebra systems include GAP and MAGMA
finally an excellent reference
