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    Midterm 3 Practice Worksheet 2: Orbits, Stabilizers, and Fixed Sets#

    This worksheet is optional practice for Midterm 3. It is not collected or graded. Work on paper, as you would on the exam, and write the method that justifies each answer. Then compare with the solutions.

    This worksheet covers group actions: the true-or-false facts about orbits and stabilizers, and the computation of orbits, stabilizers, and fixed sets of colorings. As on the exam, a symmetry moves a coloring by carrying the color at each corner to the corner ; the result is written (the lecture notes write ). The orbit of under a group is .

    The square. Label the corners of a square counterclockwise, starting at the bottom left. Its eight symmetries form the group :

    Two copies of a square with corner 1 at the bottom left, 2 at the bottom right, 3 at the top right, and 4 at the top left. The left copy has a counterclockwise arrow labeled r about the center. The right copy has four dashed mirror lines: v vertical, h horizontal, d13 through corners 1 and 3, and d24 through corners 2 and 4.
    Figure 1: The labeled square. Left: the corners and the rotation by counterclockwise. Right: the mirror lines of the reflections , , , and .#

    A coloring is written as the string . For example, .

    1. (True or false)#

    Decide whether each statement is always true, and give a one-line reason or a counterexample. In each statement, a finite group acts on a finite set .

    1. (A) For every , the stabilizer contains , and if , then .

    2. (B) For every , the fixed set is a subgroup of .

    3. (C) If , an orbit can have elements.

    4. (D) If , an orbit can have elements.

    5. (E) If , then .

    6. (F) Two different orbits can have exactly one element in common.

    7. (G) If and , then some element of is fixed by every element of .

    8. (H) If , then has elements.

    9. (I) When acts on the corners of the square, all four corners have the same stabilizer.

    10. (J) .

    2. (Colorings of the square)#

    1. (A) Let . Compute for each of the eight symmetries . Then list the orbit and the stabilizer , and check that .

    2. (B) Now use three colors, black (), white (), and red (). Let . Compute for all eight symmetries, and find and . Explain without computing why the stabilizer must be so small.

    3. (C) For black-and-white colorings, list the colorings in , in , and in . Explain how each number of colorings follows from the cycles of , , and .

    4. (D) Still with three colors, let . Compute for each of the eight symmetries . Then list and , and check that .

    5. (E) For black-and-white colorings, list the colorings in and in . Explain how the numbers of colorings follow from the cycles of and of .

    3. (Colorings of the hexagon)#

    Label the corners of a regular hexagon counterclockwise, starting at the bottom, as in the figure. Its twelve symmetries form the group : Black-and-white colorings are written as strings .

    A regular hexagon with corner 1 at the bottom and the other corners counterclockwise: 2 at the lower right, 3 at the upper right, 4 at the top, 5 at the upper left, and 6 at the lower left. A dashed vertical mirror line labeled s passes through corners 1 and 4, and a counterclockwise arrow labeled r turns about the center.
    Figure 2: The labeled hexagon, the rotation , and the mirror line of .#

    1. (A) Let . Find .

    2. (B) Use the orbit–stabilizer theorem to find the number of colorings in , and then list them.

    3. (C) For each of , , , and , find for the action on the black-and-white colorings, and explain your count using the cycles of .

    4. (D) Let . Find . Use the orbit–stabilizer theorem to find the number of colorings in , and then list them.

    Hint#

    A symmetry fixes a black-and-white coloring exactly when it sends the set of black corners to itself. A coloring is fixed by exactly when it has the same color on all the corners of each cycle of ; fixed corners are cycles of length and can have any color.

    4. (Orbit–stabilizer in other settings)#

    1. (A) The rotations of a regular tetrahedron act on its faces, and every face can be carried to every other face. Find the number of rotations that carry a given face onto itself, and use the orbit–stabilizer theorem to find the number of rotations of the tetrahedron.

    2. (B) The rotations of a cube act on its edges, and every edge can be carried to every other edge. Use the orbit–stabilizer theorem to find the number of rotations that carry a given edge onto itself, and describe them.

    3. (C) A triangular prism has two equilateral triangles as ends and three squares as sides. Its rotations act on its vertices, and every vertex can be carried to every other vertex. Show that only the identity fixes a vertex, and find the number of rotations.

    4. (D) acts on the six -element subsets of the corners of the square by . Find the orbits of this action. Find and , and check the orbit–stabilizer theorem for both.

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