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    Midterm 3 Practice Worksheet 1: Groups and Permutations#

    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 groups, subgroups, cyclic groups, and permutations: the true-or-false facts about groups and the computations with permutations and with that appear on the exam. Permutations are composed from right to left: means “first , then .” Cycle types list the cycle lengths, including fixed points, in decreasing order, and is the number of cycles of , including fixed points.

    1. (True or false)#

    Decide whether each statement is always true, and give a one-line reason or a counterexample. On the exam, true-or-false questions need no justification, but finding the reason is the best way to check your answer.

    1. (A) The group , with addition modulo , is cyclic.

    2. (B) The symmetric group is cyclic.

    3. (C) The set is a subgroup of .

    4. (D) In , .

    5. (E) A group with elements can have a subgroup with elements.

    6. (F) If an element of a group has order , then .

    7. (G) The permutation in has order .

    2. (Groups and subgroups)#

    1. (A) In , with addition modulo , list the elements of , , and . Is cyclic? List all of its generators.

    2. (B) Decide which of the following subsets are subgroups. For each subset that is not a subgroup, give a sum that leaves the subset.

    3. (C) In , compute and . Is commutative? Find the order of each of the elements of , and decide whether is cyclic.

    4. (D) A group has elements. Which of the numbers , , , can be the number of elements of a subgroup of ? Can contain an element of order ? An element of order ?

    5. (E) An element of a group has order . Which of , , , are equal to ? Write and in the form with .

    6. (F) Find the order of each permutation: , , , and .

    Hint#

    A finite subset of a group is a subgroup exactly when it is closed under the operation. By Lagrange’s theorem, the number of elements of a subgroup, and the order of an element, divide the number of elements of the group. The order of a permutation written with disjoint cycles is the least common multiple of the cycle lengths; when cycles overlap, compute the product first.

    3. (Working with a permutation)#

    Let and be the permutations in given by

    1. (A) Write in cycle notation. Give the cycle type of and .

    2. (B) Find the order of and write in cycle notation.

    3. (C) Compute the composition (first , then ) and write it in cycle notation. Give its cycle type.

    4. (D) Compute and compare it with .

    5. (E) The permutation in is written with its fixed points omitted. Give the cycle type of , , and the order of .

    4. (Another permutation)#

    Let and be the permutations in given by

    1. (A) Write in cycle notation. Give the cycle type of and .

    2. (B) Find the order of and write in cycle notation.

    3. (C) Compute the composition (first , then ) and write it in cycle notation.

    5. (Cyclic subgroups of )#

    1. (A) In , with addition modulo , list the elements of the cyclic subgroups and , and give the orders of the elements and .

    2. (B) Is a generator of ? Is ? Explain, and list all generators of .

    3. (C) In , list the elements of . Is a generator of ? List all generators of .

    4. (D) In , list the elements of the cyclic subgroup and give the order of the element . Is a generator of ? Explain.

    Hint#

    In the “powers” of are its multiples reduced modulo . Keep adding until you reach ; the number of elements you listed, including , is the order of .

    6. (Symmetries of the regular pentagon)#

    Label the corners of a regular pentagon counterclockwise, starting at the bottom, as in the figure. Let be the rotation by counterclockwise, so (with meaning ), and let be the reflection across the axis through corner and the center.

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

    1. (A) Write , , , , and as permutations of the corners in cycle notation.

    2. (B) Compute and , and check that they are equal. Is ?

    3. (C) Find the order and of every element of . Is cyclic?

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