#### Date of Graduation

Spring 2012

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Kishor Shah

#### Keywords

Group Theory, 3x3x3 Rubik's Cube, 5x5x5 Rubik's Cube, Slice Group, Semidirect Products

#### Subject Categories

Mathematics

#### Abstract

We study some of the group theory underlying the 3x3x3 Rubik's Cube. We prove the well-known Fundamental Theorem of Cubology for the 3x3x3 Rubik's Cube, filling in various details. We introduce and study four interesting subgroups of the 3x3x3 Rubik's Cube Group. In addition, we study the well-known Slice Group, again filling in various details. The group theory of the 5x5x5 Rubik's Cube is not well-developed in mathematical literature. We introduce and study some group theory for the 5x5x5 Rubik's Cube, which, to our knowledge, may be new: we state and prove a Fundamental Theorem of Cubology for the 5x5x5 Rubik's Cube. We also introduce and study two groups which arise from the 5x5x5 Rubik's Cube: the Full Group and the Visual Group. We then examine the relationship between these two groups. We relate the 5x5x5 Rubik's Cube Group to the 3x3x3 Rubik's Cube Group. We conclude our thesis by making some brief comments on the (2n+1)x(2n+1)x(2n+1) Rubik's Cube.

#### Copyright

© Kristen Michelle Howell

#### Recommended Citation

Howell, Kristen Michelle, "Group Theory Underlying the 3X3X3 and 5X5X5 Rubik's Cubes" (2012). *MSU Graduate Theses*. 1640.

http://bearworks.missouristate.edu/theses/1640