I'm interested in the connection between Rubik's cube and group
theory. I want to find resources about illustrating basic concepts in
group theory and composing moves to achieve certain results on the
cube.

There are a number of great resources describing a relationship
between the game Rubik's Cube and the mathematical studies of Group
Theory.

In almost all descriptions or courses about Group Theory, the Rubik's
Cube is used as an example of a permutation puzzle.

"There is math in Rubik's Cube that you haven't learned yet. The math
in Rubik's Cube is a branch of mathematics called Group Theory.
Instead
of multiplying or adding numbers, you rotate faces of the cube, and 
this kind of math helps to describe what happens when you do several 
rotations in a row.

For instance, in regular math, 3 times 2 is 6 and 2 times 3 is also 6.
But in the "group" that is Rubik's Cube (the set of rules that it 
follows), turning the top counterclockwise and then turning the front
clockwise is NOT the same as doing these two "operations" in the 
reverse order. We say that the group is not "commutative"."
http://mathforum.org/dr.math/problems/smith12.3.98.html

Since you ask for resources instead of concepts, here are some to get
started with.

"Introduction to Group Theory" is a 14 chapter discourse on Group
Theory including 3 chapters on Rubik's Cube analogy and implications.
http://members.tripod.com/~dogschool/

Marlboro College course notes and assignments on "Group Theory &
Rubik's Cube"
http://www.marlboro.edu/~mahoney/courses/Spr00/rubik.html#defs

Lecture Notes on Mathematics of the Rubik's Cube
http://web.usna.navy.mil/~wdj/rubik_nts.htm


Additional information:

Rubik's Cube/Group Theory Site Links
http://web.idirect.com/~cubeman/cubeman.html

Google Web Directory: Group Theory
http://directory.google.com/Top/Science/Math/Algebra/Group_Theory/

DMOZ: Open Directory: Group Theory
http://dmoz.org/Science/Math/Algebra/Group_Theory/

General information on Rubik's Cube including "best algorithms" and
visualization tools
http://www.site.uottawa.ca/~istvan/rubik/rubikPage.html

Rubik's Cube for Windows
http://www.winsite.com/bin/Info?500000031227


Search terms used:

rubik cube group theory
://www.google.com/search?q=rubik+cube+group+theory