Visual Group Theory (Nathan Carter)

18 flashcards · Mathematics · by lakesare

1. What is an interesting difference between: Rubik's cube puzzle self-made rectangle puzzle

   Rubik's cube internalizes the puzzle - we don't need external reference!

2. [Notation] How do we read this?

   { r, f } is a generator set of S3.

3. [Cayley Diagram] In a Cayley diagram - what are: states VS arrows? Cayley diagram of the Klein 4-group

   states - state (can be described as different combinations of elements!) arrows - group elements

4. Why do groups relate to symmetry of objects?

   Groups == { ways to rearrange a string } == From Cayley's theorem.{ symmetries } Because symmetry can be described using rearrangement of object parts.

5. Name 2 steps to find symmetries of an object.

   1. Give names to the parts of the object which are similar. Identify all the parts of the object that are similar, and give each such part a different number. (The rectangle had four alike corners, so we numbered them. We could have instead numbered the four sides, because the left is the same as the right and the top is the same as the bottom.)2. Consider actions you can perform with your hands. Consider the actions that you can perform with your hands that may rearrange the numbered parts, bu…

6. Visually imagine a frieze.

   repeatsrepeatsrepeats A frieze pattern, which can be extended indefinitely to the left or right. In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group. The cornices of some buildings are sculpted in a one-dimensional repeating pattern…

7. [Cayley Diagram] Name 2 reasons we only use generators as arrows to connect Cayley diagram?

   they reach all nodes don't clutter the diagram

8. Visually imagine how Galois showed the unsolvability of the quintic.

   polynomial = 0 solutions = { ... } // This is a group! By examining this group, we can tell how easy or hard it is to solve the equation.

9. [Cayley Table] How to read the Cayley Table without headers?

10. [Cayley Diagram VS Cycle Graph] What's the difference between: Cayley Diagram Cycle Graph

    Cycle Graph is a Cayley Diagram (of a cyclic group!), but, by convention, without arrowheads.

11. [Recognizing groups] How do cyclic groups look in a Cayley Table?

    Like a rainbow. Modular addition has a nice visual effect on multiplication tables; each row is different from the row above it only by being cycled one cell to the left. In colored multiplication tables, this extends the color spectrum in the top row diagonally through the table, creating a rainbow effect. Cayley table of a group C15

12. [Recognizing groups] How do abelian groups look in a Cayley Table?

    A mirror across the diagonal. The equation ab = ba, which defines abelian groups, causes their multiplication tables to be symmetric across the diagonal shown here

13. What's a dihedral group?

    { symmetries of a regular polygon } The symmetry group of a snowflake is D6, a dihedral symmetry, the same as for a regular hexagon. Regular polygons

14. [Dihedral groups] Name 2 things n might refer to in a dihedral group Dn.

    all symmetries of an n-gon (group of order 2n) dihedral group of order n (group of order n)

15. [Recognizing groups] How do dihedral groups look in a Cayley Diagram?

    Cayley diagrams for the dihedral groups D3, D5, and Dn. The outer ring in each of these diagrams is the orbit of r , and is a copy of the cyclic group Cn. The inner ring is a cycle of the same size, but pointing counterclockwise instead of clockwise. To see the reason for this reversal, imagine yourself holding a triangle in the initial configuration shown on the left of Figure 5.16. A friend stands facing you, and you hold the triangle up between the two of you, so that you see the front face…

16. [Recognizing groups] How can you see that this is a dihedral group? Cayley Table for the dihedral group D5. Cayley Diagram for the dihedral group D5.

    left top: only interactions of the outer ring This separation of the elements of Dn into two categories - the inner and outer rings - is also visible in multiplication tables. It is not difficult to mentally divide this table into quarters by splitting it in half horizontally and again vertically; in fact, the colors of the cells encourage you to do so.This is because the headings in the table have been organized with those from the Cayley diagram's outer ring listed first (e through r4) and it…

17. Visually imagine taking a quotient of the group. Cayley Table for the dihedral group D5.

    Shrink the group. Multiplication table for D5 with annotation that clusters elements according to whether they involve a flip. The resulting two-by-two pattern matches the multiplication table for the group of order two, shown on the right.

18. Name 2 ways to generate: the alternating group from the symmetric group.

    square each el The left column shows all the elements of S3 as permutations, and the right column shows all their corresponding squares. Because four of the six elements square to the identity, the right column contains only three distinct elements. Those three elements comprise the group An.only take even permutations