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    Midterm 3 Practice Worksheet 2: Solutions#

    These are the solutions to Practice Worksheet 2. The write-ups model the amount of explanation expected on the exam. A symmetry moves a coloring by carrying the color at each corner to the corner ; the result is .

    True or false#

    (A) True. , so . If and , then . (These two facts make a subgroup of .)

    (B) False. is a set of elements of , not of . For example, when acts on colorings of the square, is a set of colorings.

    (C) False. By the orbit–stabilizer theorem, , so every orbit size divides , and does not.

    (D) True. For example, has elements and acts on the corners of the regular pentagon with a single orbit of corners.

    (E) True. Write . Then and for every , so each orbit is contained in the other.

    (F) False. By (E), two orbits with a common element are both equal to . Different orbits are disjoint.

    (G) True. The orbit sizes divide , so they are , , , or , and they add up to . Since is odd, some orbit has odd size, and the only odd choice is . An orbit means for every .

    (H) True. .

    (I) False. , while . The stabilizers of corners in one orbit have the same size, but they need not be equal.

    (J) True. for every .

    Colorings of the square#

    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) For each , the color at corner moves to corner . For example, moves the colors of corners (B, B, B, W) to corners , which gives BBWB. All eight results: The different results form the orbit the four colorings with exactly one white corner. The results equal to occur for and , so : these are the symmetries that keep the white corner in place. Check: .

    (B) In the same way: All eight colorings are different, so consists of these colorings and .

    Without computing: a symmetry fixing must keep the only red corner and the only white corner in place. These corners are adjacent, and a symmetry of the square is determined by where it sends corner and one neighbor of . So only fixes , and the orbit has colorings.

    (C) A coloring is fixed by exactly when it has one color on each cycle of , so .

    • has cycles: corners agree and corners agree. , colorings.

    • has cycle: all corners agree. , colorings.

    • has cycles: corners agree, and corners and are free. colorings. The fixed corners and are cycles too; forgetting them would give the wrong count .

    (D) The color at corner moves to corner . For example, moves the colors of corners to corners , which gives WRBR. All eight results: So and : the reflection exchanges the two red corners and fixes the black corner and the white corner . Check: .

    (E) As in (C), . has cycles: corners and agree, and corners and are free. There are fixed colorings: has cycles: corners agree and corners agree. There are fixed colorings:

    Colorings of the hexagon#

    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 .#

    (A) A symmetry fixes exactly when it sends the set of white corners to itself. Check each element:

    • rotations: fixes everything; exchanges ; , , , send to , , , , none of which is white.

    • reflections: exchanges , and fixes and ; the others send elsewhere ( to , to , to , to ).

    So .

    (B) By the orbit–stabilizer theorem, . The white corners are opposite, and symmetries send opposite corners to opposite corners, so the orbit consists of the colorings whose two white corners are opposite: (For instance, and .)

    (C) .

    • : cycles, so fixed colorings (the odd corners agree and the even corners agree).

    • : cycles, so .

    • : cycles, including the fixed corners and , so .

    • : cycles, so .

    (D) A symmetry fixes exactly when it sends the black corners to themselves. The rotations , , move every corner an even number of steps, so they do; , , move every corner an odd number of steps and send to . Among the reflections, , , and send to itself, while , , send corner to the even corners , , . Hence and by the orbit–stabilizer theorem : where .

    Orbit–stabilizer in other settings#

    (A) A rotation that carries a face onto itself keeps the center of that face and the center of the tetrahedron in place, so it turns the tetrahedron about the line through them, which also passes through the opposite vertex. The rotations about this axis are by , , and , so the stabilizer has elements. The orbit of a face consists of all faces, and the tetrahedron has rotations.

    (B) The orbit of an edge has elements, so . The two rotations are the identity and the half turn about the axis through the midpoint of the edge and the midpoint of the opposite edge, which exchanges the two ends of the edge.

    (C) Each vertex lies on exactly one of the two triangles. A half turn about a horizontal axis exchanges the two triangles, so it moves every vertex. The other rotations turn the prism about the vertical axis through the centers of the triangles; the rotations by and move every vertex. So only the identity fixes a vertex, and the prism has rotations: the identity, two turns about the vertical axis, and three half turns.

    (D) Symmetries preserve adjacency, so they send sides to sides and diagonals to diagonals; the rotations carry any side to any other side, and any diagonal to the other one. So there are two orbits: the sides , and the diagonals .

    : exchanges and , and every other nonidentity symmetry sends or outside . Check: .

    : fixes and , while and exchange them; , , , send the diagonal to . Check: .

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