How many times do you need to shuffle a deck of cards?

This is a famous problem solved using group thoery in a series of papers:
Diaconis P. (1988)Group representations in probability and statisticsIMS Lecture Notes IMS, Hayward California.

Diaconis P., Mc Grath M. and Pitman J. (1994) Riffle Shuffles, Cycles and Descents. Combinatorica 66 379-386.

For a hypertext explanation try this site: http://www.geom.umn.edu/docs/snell/chance/teaching_aids/card_shuffling/Mann/Mann.html

The exposition I sketched is well developped there and in: http://www.geom.umn.edu/docs/snell/chance/teaching_aids/card_shuffling/shuf/shuf.html