**math.GR Group Theory**

- Group theory provides methods for understanding the Rubik's cube, and for [generating algorithms](http://www.math.toronto.edu/~drorbn/Talks/Mathcamp-0907/NCGE.html) for solving the cube remarkably quickly from any state the cube may be in.
- Groups find various applications in chemistry, eg. in the study of crystal structures and spectroscopy.
- Cryptography - various hard algorithmic problems about groups are used to design crypto-systems.
- Groups of symmetries are used to reduce the dimension of parameter spaces in engineering models to make model verification more tractable.
- Potentially fast matrix multiplication; see [this MO question][1].
- Card tricks that don't work by sleight of hand, but via the arrangements of the cards. e.g. Sim Sala Bim, see [this site for a description][2]. If you think about it, the symmetric group explains the trick and shows you how you to extend it past three piles of seven cards, but to N piles of M cards.  


  [1]: http://mathoverflow.net/questions/34173/fast-matrix-multiplication
  [2]: http://www.angelfire.com/blog/card-magic/21ct.html