User:Hgy0407/sandbox

Groups, Graphs and Geometry
Question: How many ways are there to get the same figure of a cube by rotating it?

Symmetry
objectives: rigid motion, symmetries in a plane

Definition: rigid motion : $$ m : \textbf{R}^2 \to \textbf{R}^2 $$ such that for any point p and q in a plane $$ \textbf{R}^2 $$, we have $$\left\vert m(p)-m(q) \right\vert = \left\vert p-q \right\vert $$.

Definition: symmetry

Examples: identity, translations, rotations, reflections, glide reflections. Those can be classified into two classes: orientation-preserving and orientation reversing.

Since it takes too much effort on looking at each subsets of a plane individually, we generalize the concept of a symmetry.

Group
objectives: group, subgroup, isomorphism, homomorphism, automorphism, product group

Definition: group

Symmetry groups
objectives: group action, the counting formula

Group presentations
objectives: group presentation, free group, cayley graph