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Abstract Algebra, I don't understand what my HW question is asking!

  1. Jul 7, 2009 #1

    gmn

    User Avatar

    Abstract Algebra- VERY SIMPLE but I don't understand what my HW question is asking!

    1. The problem statement, all variables and given/known data


    Hi. I am having trouble simply understanding what the question is here:

    (6) let w = (1 2 3 4 5 6 7 8 9 10 11 12 13 14). For which integers i is w^i a 14-cycle?



    Here is a link to the assignment if you would prefer to read it there, it's #6:

    http://math.berkeley.edu/~rdore/113/hw3.pdf [Broken]


    2. Relevant equations

    SX ={set of permutations on 1,......,X}

    definition of a cycle of length k, or k-cycle:

    A permutation o in SX is a cycle of length k if there exist elements
    a1; a2; : : : ; ak in X such that
    o(a1) = a2
    o(a2) = a3
    ...
    o(ak) = a1
    and o(x) = x for all other elements x in X. We will write (a1 a2 ....... ak) to
    denote the cycle o. Cycles are the building blocks of all permutations.


    SO, I'm pretty sure I'm not confused about the following information right:

    *so in w, take id = (1 2 3 4 5 6 7 8 9 10 11 12 13 14)
    and w(id) = (2 3 4 5 6 7 8 9 10 11 12 13 14 1) [1 goes to 2, 2 goes to 3, etc.]

    w^3= w(w(w(id):

    w(w(w(1)))= w(w(2))= w(3) = 4
    www(2) = ww(3)=w(4)=5
    .
    .
    .
    www(14) = ww(1) = w(2) = 3
    so w^3=(4 5 6 7 8 9 10 11 12 13 14 1 2 3)
    .
    .
    .

    w^i = w(w(w(.........(w(id))))) w composed with w i times.


    3. The attempt at a solution

    I have

    w^2 = w(w(id))= w(2 3 4 5 6 7 8 9 10 11 12 13 14 1) = ( 3 4 5 6 7 8 9 10 11 12 13 14 1 2).

    w^3 = ( 4 5 6 7 8 9 10 11 12 13 14 1 2 3).
    w^4 = ( 5 6 7 8 9 10 11 12 13 14 1 2 3 4).
    .
    .
    .
    w^13 = id = ( 14 1 2 3 4 5 6 7 8 9 10 11 12 13).
    w^14 = w = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ).
    w^15= w^2
    .
    .
    .
    w^i = w ^ imod13

    all 14-length cycles. ???? Is that all? I dunno. I'm confused.

    What I'm having trouble with is the question of whether or not there are any i's for which w^i is not a 14 cycle? I can't really extract any relevent information with which to make a substantial proof. Maybe I'm not getting something? It just seems to me like it would go on and on in a loop forever, as there does not seem to be any disjoint cycles to be extracted or anything that should change when taking w with itself.
    thanks!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 8, 2009 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi gmn! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    Noooo :cry:

    w sends 1 to 2 and 2 to 3, so w2 sends 1 to 3, 3 to 5, …

    so w2 = (1 ? ? ? …)(? ? …) :smile:
     
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