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    Homework 7#

    Due: Thursday, October 29, 2026 23:59
    Submit: Gradescope, through Blackboard’s assignment area.

    Please write your homework by hand. Start each exercise on a new page, scan your work to a legible PDF, and clearly mark the pages associated with every exercise. Compose permutations from right to left, as in the lecture notes: means “first , then .”

    1. (Words in )#

    Points: 10

    Recall the presentation of the symmetries of the regular hexagon, with corners labeled counterclockwise. Use only the relations, not tables, except where a part asks you to check an answer by computing where corners go.

    1. (A) 2 points. Prove that for every integer by applying the relations.

    2. (B) 5 points. Write each of the following words in the normal form with and : Recall that every element of is either a rotation or one of the six elements , each of which is a reflection of the hexagon across a line through its center. For each word, say whether it is a rotation or a reflection. If it is a rotation, give its angle. If it is a reflection, draw the hexagon with its corners labeled, draw the axis of reflection, and state which corners, if any, lie on the axis. Check your answer by computing where the element sends each corner.

    3. (C) 3 points. Show that for every . Then find the order of each of the elements of , and check that every order divides .

    2. (Computing in )#

    Points: 10

    Let

    1. (A) 2 points. Write in cycle notation, and write as a two-row table.

    2. (B) 2 points. For each of and , list the lengths of all of its cycles, including cycles of length , and state the total number of cycles. Then find the order of and the order of , that is, the smallest positive integer with (respectively ), and explain how you obtain it from the cycle lengths.

    3. (C) 2 points. Find , both in cycle notation and as a table.

    4. (D) 2 points. Compute and in cycle notation. Are they equal? List the lengths of the cycles of and of . Do the two products have the same number of cycles of each length?

    5. (E) 2 points. Compute in cycle notation without computing compositions.

    Need a free hint?#

    For (E), divide by the order of and keep only the remainder. Then handle each cycle of separately: a power of a permutation is the composition of the same power of each of its cycles.

    3. (The cyclic group )#

    Points: 10

    Recall that in the operation is addition modulo , so the -th power of is the remainder of on division by .

    1. (A) 3 points. Find the order of each of the elements of .

    2. (B) 3 points. For , the cyclic subgroup is the set of all powers of . In these are the elements reached by starting from and adding over and over, always taking the remainder on division by . For example, in the smaller group , adding repeatedly gives , so , and adding repeatedly gives , so .

      List the elements of for every . How many different sets appear in your list? Explain how Lagrange’s theorem predicts the possible numbers of elements of these sets.

    3. (C) 2 points. An element generates if every element of is a power of , that is, if is all of : adding over and over eventually reaches every one of . In the example in (B), generates , but does not.

      Use your list from (B) to find all elements that generate . State a rule, in terms of and , that predicts which are generators.

    4. (D) 2 points. A group is cyclic if some element generates it, that is, every element of is a power of . For example, is cyclic, because generates it. Decide whether each of , , and is cyclic, and justify your answer. Use Lagrange’s theorem to explain why has no subgroup with elements.

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