I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem.Is there any way of visualizing these things?Please suggest some good reference book/text also which discusses these things and also the motivation/idea behind different theorems and concepts.
One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.
To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.