In mathematics, symmetry has more than just a visual or geometric quality. Mathematicians comprehend symmetries as motions—motions whose interactions and overall structure give rise to an important mathematical concept called a "group." This unit explores Group Theory, the mathematical quantification of symmetry, which is key to understanding how to remove structure from (i.e., shuffle) a deck of cards or to fathom structure in a crystal.

Unit Goals

• Symmetry, in a mathematical sense, is a transformation that leaves an object invariant.
• Some symmetries are understood as geometric motions.
• Some symmetries are understood as algebraic operations.
• When symmetries are combined, another symmetry is the result.
• Symmetries form what is known as a group, which allows mathematicians to perform a sort of "arithmetic" with things that are not numbers.
• A "frieze group" is the set of symmetries of an infinite frieze (a frieze is a pattern that repeats along a line — i.e., in one dimension).
• All frieze groups fall into one of seven general types. 
• A "wallpaper group" is the set of symmetries of an infinite wallpaper pattern — that is, a pattern that "repeats" in two dimensions.
• Wallpaper groups fall into one of 17 general categories.
• All groups of symmetries can be expressed as permutations, which also form groups.
• The ways in which a polynomial's roots behave under a certain collection of permutations determine whether or not the roots have a simple algebraic formula (such as the quadratic formula) in terms of the coefficients of the polynomial.
• Every conserved physical quantity is based on a continuous symmetry.